MAIN Let G(q)=_2(m(q)) be an exponential generating function, where Li₂ is the polylogarithm of order 2, _2(z)=^\infty {k^2} and m(q) is the inverse elliptic nome which can be expressed through the Dedakind eta function as m(q)={2})^{8}\eta(2\tau)^{16}}{\eta(\tau)^{24}} where q = eiπτ or by Jacobi theta functions m(q)=\left({\theta_3(0,q)}\right)^4 where \theta_2(0,q)=2^\infty q^{(n+1/2)^2}\\ \theta_3(0,q)=1+2^\infty q^{n^2} giving explicitly G(x)=^\infty {k^2}\left(^\infty x^{(n+1/2)^2}}{1+2^\infty x^{n^2}}\right)^{4k}=^\infty {k!} if we consider the sequence of coefficients ak associated with G(x), modulo 1, or the fractional part of the coefficients, frac(ak) we gain the following sequence 0,0,0,{3},0,{5},0,{7},0,0,0,{11},0,{13},0,0,0,{17},0,{19},0,0,0,{23},0,0,0,0,0,{19},0,{31},0,0,0,0,0,{37},0,0,0,{41},\cdots we see the primes in the denominator in positions where the power of x is a prime. We also note that so far, the numerators are always less than the denominator (obviously), but count, succesively upwards, producing monotonically increasing subsequences. The prime only parts continue {3},{5},{7},{11},{13},{17},{19},{23},{29},{31},{37},{41},{43},{47},{53},{59},{61},{67},{71},{73},{79},{83},{89},{97}, After closer inspection, we see the numerators from the point 1, 3, 7, 13, 15, 21, 25, 27, 31, 37, 43, 45, 51, 55, 57, ... take the form prime(k)−16, the numerators before this take the form 2 ⋅ prime(k)−16, for 6, 10, 3 ⋅ prime(k)−16 for 5, 4 ⋅ prime(k)−16 for 4 and 6 ⋅ prime(k)−16 for the first numerator 2. It is likely then that for the rest of the numbers this pattern continues. This then gives for the coefficient ak of G(x), with k > 6, (a_k)={k}, \; k\in We find that if we take the original coefficients ak, and subtract this fractional part in general \delta_k=a_k-{k} for numbers m which cannot be written as a sum of at least three consecutive positive integers, δm is an integer (empirical). A111774 “ Numbers that can be written as a sum of at least three consecutive positive integers.” apart from odd primes, numbers which cannot are powers of two. OTHER We find a similar relationship with G_2(x)=_2\left({(1-x)^2\left(1-{x-1}\right)^2}\right)=^\infty {k!} where bk seem to follow for k > 2 (b_k)={k}, \; k\in GENERATING FUNCTION FOR FRACTIONAL PART We see the Generating function for n/2 is {2(x-1)^2} but the generating function for the fractional part of n/2, which is (n mod2)/2, is given by {2(x^2-1)} the property described is associated with the polylog, and we seen that the fractional part of _2(2x)=^\infty {k!} gives (c_k)={k}, \; k\in\\ 0, \; this means \left({k^2}\right) = {k}, \; k\in\\ 0,\; or \left({k}\right)= {k}, \; k\in\\ 0,\; we also see that \left({k}\right)= {k}, \; k\in\\ {2}, 4\\ 0,\;

_Article details_ Software metapaper, structured for the Journal of Open Research Software (JORS). Online version (Authorea) can be found at https://www.authorea.com/users/71114/articles/122402/_show_article Github (software repository): github.com/phockett/ePSproc Figshare repository (manuscript & source files): DOI: 10.6084/m9.figshare.3545639 10/08/16 - This fork for review. 12/11/16 - arXiv version uploaded, 1611.04043 03/09/19 - Finally working on a python version, see Github pages for updates. Full documentation now on on Read the Docs.

ABSTRACT I note down a curious relationship in the sequence OEIS A245479, which displays a novel property. The sequence is titled “Numbers n such that the n-th cyclotomic polynomial has a root mod 7.”, and is easily described by a generating function and explicit formula. The sequence also appears to display a product recurrence relation in the following manner. If n = 2α + 2β + 2γ + ⋯, the binary polynomial for n, then the (n + 1)th term of the sequence a(n + 1) is given by a(n+1)=a(2^\alpha+1)\cdot a(2^\beta+1)\cdot a(2^\gamma+1)\cdot \cdots where 2α, 2β, 2γ... are necessarily less than or equal to n. Thus each higher term in the sequence can be described by a product of the lower terms. MAIN The generating function for our sequence (indexed from 1) with terms a(n) is A(x)=^\infty a(k)x^k we may also write this as a formal product series A(x)=^\infty (1+b(k)x^k) If we do this for A245479, 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294, ⋯ we find the coefficients b(n) to be 2, 3, 0, 7, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 2401, ⋯ this means we can write x+2x^2+3x^3+6x^4+7x^5+14x^6+\cdots = x(1+2x)(1+3x^2)(1+7x^4)(1+49x^8)(1+2401x^{16})\cdots we can notice that only the powers of 2 appear in x on the right hand side, and the coefficients of these powers are the (2n + 1)th terms of a(n). This means we may write ^\infty a(k)x^k=x^\infty (1+a(2^k+1)x^{2^k}) if this is true it will immediatly follow that a(n+1)=a(2^\alpha+1)\cdot a(2^\beta+1)\cdot a(2^\gamma+1)\cdot \cdots with n=2^\alpha+2^\beta+2^\gamma+\cdots for example let n = 31 31=2^0+2^1+2^2+2^3+2^4 then we have a(32)=a(2)a(3)a(5)a(9)a(17) inserting the numbers 4941258=2\cdot3\cdot7\cdot49\cdot2401 which is true. This can also be written 4941258=2\cdot3\cdot7\cdot7^2\cdot7^4 CONCEPT Like a logarithm satisfies f(ab)=f(a)+f(b) and an exponential f(a+b)=f(a)f(b) we the have a function which over the integers satisfies f\left(1+} 2^b\right)=}f(1+2^b) what is this function? We may take the inverse Z-Transform of the generating function from OEIS, with a reciprocated input A\left({x}\right)=-{x}\left({x}+1\right){x^2}+1}{{x^4}-1} f(n)^{-1}(A)= \cdots we see it has complex terms when extended to non-integer values of n. We see that in general this relationship does not hold for non-integer values of n, as f(+^3+1)\ne f(+1)f(^3+1) Let us call this function bmf(n). Then we may begin to unravel it’s power. USES If an integer p = m + 1 is an odd prime it must have a binary term 2⁰. Then (p+1)=(2)(m+1) but m = p − 1 so (p+1)=(2)(p) then (p+1)}{(p)}=2 we see from inspection that (n+1)}{(n)}=2 for all odd n. That’s a start. If we consider Mersenne primes, whose binary representation may not contain any 0, we note that for a Mersenne prime p = 2k − 1 (p+1)=(2)(3)(5)\cdots\left({2} +1\right)\\ (p+1)=(2)(3)(5)\cdots\left(2^{k-1}+1\right)

ABSTRACT We are all aware of the concept of integers 1, 2, 3, 4, 5, 6, ⋯. Which all have the nice property of a unique representation of a product of powers of primes. We can write numbers by their divisibility by a prime, or just the powers of the primes in this product for example 2=1\\ 3=0,1\\ 4=2,0\\ 5=0,0,1\\ 6=1,1\\ 7=0,0,0,1 where we see each new digit is the next prime. We can have powers greater than one etc. Any expression we can write with any number of commas, and any powers will then be a real, tangible integer. Now, instead of prime factors, let each new digit in such an expression be a positive divisor, and the “power” (it’s not really a power any more) is the number of times that divisor can be applied 1=1\\ 2=1,1\\ 3=1,0,1\\ 4=1,2,0,1\\ 5=1,0,0,0,1\\ 6=1,1,1,0,0,1 this creates the standard divisor map pattern but there is a non-standard 2 in the representation of 4 for example. Now, we can realised that only _certain_ expressions in this format constitute actual integers. If we write 1, 0, 0, 1, this would then imply a number which can be divided by 4 once, and by 1, but not by 2 at all! Obviously this would be a contradiction if it were an integer, so it is a fictional integer that does not exist. However, we can now explore the concepts behind some of these numbers. MAIN Let a string of integers seperated by commmas as above be called an “aggregate”. We can tabulate the first few examples of aggregates below with some arbitrary labels a=1\\ b=0,1\\ c=1,1\\ d=0,0,1\\ e=1,2\\ f=1,3 we can see that a = 1, the normal unity we are used to, b however, is a fictional integer, it is divisible by 2 but not by 1. c is an integer and c = 2. d is a fictional form of 3 which is not divisible by 1, e is a fictional form of 2, which is divisible by 2 twice, but not by 4. f is divisible by 2 three times but not by 4 nor by 8. We can define a product, of aggregates, which is simply a sum of constituents. for example 0,1 \cdot 1,1,0,1 = 1,2,0,1 = 4 above we can see that two fictional integers, can form an integer under this product. One question is the validity of a digit to represent 1. Fictional integers which can not be divided by 1 are in a way more strange than fictional integers that can. We can define a span of an aggregate, as the number of digits written, which is also the largest number that aggregate is divisible by \; a,b,c,\cdots,z = 24 we can also define the fill of an aggregate, which is the sum of all the entries \; a,b,c,d = a+b+c+d we can write any ’positive’ aggregate, where positive is that each digit is positive or zero, as the product of one or more fill-1 aggregates, for example [a,b,c,d] = [a] \cdot [0,b] \cdot [0,0,c] \cdot [0,0,0,d] Theorem 1, there is only one fill-1, span-n aggregate, for each n. Proof, there is only one way to arrange m − 1 zeroes and one 1 such that the 1 is in the mth position. Theorem 2, all prime numbers are the product of the unit aggregate and the fictional fill-1 prime aggregate. Proof trivial by above theorem. for example 7=[1,0,0,0,0,0,1] =[1]\cdot[0,0,0,0,0,0,1] Division is the inverse of the product, so one may think ⋅, 6/3=[1,1,1,0,0,1]/[1,0,1]=[0,1,0,0,0,1] and we see the aggregate division of two integers can yield a fictional integer. This poses a problem. In the original notation we have placed a 1 in the 1’s column of the aggregate, but all numbers are divisible by 1 as many times as needed. So really they should have an ugly ∞ in the 1’s column. This mans when dividing we should always re-add a one if one was present in the first place, or altogether ignore the 1′s column of the aggregates. MORE OPERATIONS We may create a number of different operations for such numbers, for example substitution/composition [1,0,2,3]\#[1,0]\to[[1,0],[0,0],[2,0],[3,0]]=[1,0,0,0,2,0,3,0] a number which is divisible by 1, 5 twice and by 7 three times, this may seem useless, however, note that [0,1]\#[0,1]=[0,0,0,1] [0,0,1]\#[0,0,1]=[0,0,0,0,0,0,0,0,1] [0,1]\#[0,0,1]=[0,0,0,0,0,1] [0,0,1]\#[0,1]=[0,0,0,0,0,1] so we see, we may write composite m, fill-1, span-m aggregates as # substitutions of their factors which appear to commute. Then we may write 4=[1,2,0,1]=[1]\cdot[0,1]\cdot[0,1]\cdot[0,0,0,1] but [0,0,0,1]=[0,1]\#[0,1] 4=[1,2,0,1]=[1]\cdot[0,1]\cdot[0,1]\cdot([0,1]\#[0,1]) 2=[1,1]=[1]\cdot[0,1]\\ 2\times2=([1]\cdot[0,1])\times([1]\cdot[0,1])=[1]\cdot[0,1]\cdot[0,1]\cdot([0,1]\#[0,1]) if we accept the rules [1]\cdot[1]=[1]\\ [1]\#[1]=[1] then this reads a\times a= a\cdot a \cdot(a\# a) for 9 we may try 3\times 3= ([1]\cdot[0,0,1])\times([1]\cdot[0,0,1])= [1]\cdot[0,0,1]\cdot[0,0,1]\cdot([0,0,1]\#[0,0,1])=[1,0,2,0,0,0,0,0,1]=9 so we may represent the squaring of integers under this method, in fact it appears to work for composites with two factors 3\times 2= ([1][0,0,1])\times([1][0,1])= [1][0,1][0,0,1]\cdot([0,1]\#[0,0,1])=[1][0,1][0,0,1][0,0,0,0,0,1]=6 long winded, but consistent on the priviso we write the numbers explicity in their prime-product form! Then try 2\times6 = ([1][0,1])\times([1][0,1][0,0,1][0,0,0,0,0,1])= [1][0,1][0,1][0,0,1]([0,1]\#[0,1])([0,1]\#[0,0,1])([0,1]\#[0,0,0,0,0,1]) which makes less sense, it would appear we take all but the last divisor in the sequence as a product with [1] and then take the # product with all divisors. As a final check we try two composites in a product 6\times10= [1][0,1][0,0,1][0,0,0,0,0,1]\times[1][0,1][0,0,0,0,1] = [1\#1][1\#2][2\#1][3\#1][2\#2][1\#5][6\#1][2\#5][6\#2][3\#5][2\#2\#5][6\#5][6\#2\#5] MAGNITUDE We can also define a magnitude, which is a weighted fill. For an aggregate [a,b,c,d]= a+2b+3c+4d this gives us a way to rank aggregates, we can then rank first by magnitude, then by span, then by fill. We can tabulate some below & & & & & \\ (0) & 0 & 0 & 0 & y & n\\ (1) & 1 & 1 & 1 & y & n\\ (2) & 2 & 1 & 2 \\ (0,1) & 2 & 2 & 1 \\ (3) & 3 & 1 & 3 \\ (1,1) & 3 & 2 & 2 & y & y\\ (0,0,1) & 3 & 3 & 1 \\ (4) & 4 & 1 & 4 \\ (0,2) & 4 & 2 & 2 \\ (2,1) & 4 & 2 & 3 \\ (1,0,1) & 4 & 3 & 2 & y & y\\ (0,0,0,1) & 4 & 4 & 1 \\ (5) & 5 & 1 & 5 \\ (1,2) & 5 & 2 & 3 \\ (3,1) & 5 & 2 & 4 \\ (0,1,1) & 5 & 3 & 2 \\ (2,0,1) & 5 & 3 & 3 \\ (1,0,0,1) & 5 & 4 & 2 \\ (0,0,0,0,1) & 5 & 5 & 1 \\ (6) &6 &1 &6 \\ (0,3) &6 &2 &3 \\ (2,2) &6 &2 &4 \\ (4,1) &6 &2 &5 \\ (0,0,2) &6 &3 &2 \\ (1,1,1) &6 &3 &3 \\ (3,0,1) &6 &3 &4 \\ (0,1,0,1) &6 &4 &2 \\ (2,0,0,1) &6 &4 &3 \\ (1,0,0,0,1) &6 &5 &2 & y & y\\ (0,0,0,0,0,1) &6 &6 &1 \\ (7) & 7 &1 &7\\ (1,3) & 7 &2 &4\\ (3,2) & 7 &2 &5\\ (5,1) & 7 &2 &6\\ (0,2,1) & 7 &3 &3\\ (1,0,2) & 7 &3 &3\\ (2,1,1) & 7 &3 &4\\ (4,0,1) & 7 &3 &5\\ (0,0,1,1) & 7 &4 &2\\ (1,1,0,1) & 7 &4 &3\\ (3,0,0,1) & 7 &4 &4\\ (0,1,0,0,1) & 7 &5 &2\\ (2,0,0,0,1) & 7 &5 &3\\ (1,0,0,0,0,1) & 7 &6 &2\\ (0,0,0,0,0,0,1) & 7 &7 &1 \\ (8) &8 &1 &8 \\ (0,4) &8 &2 &4\\ (2,3) &8 &2 &5\\ (4,2) &8 &2 &6\\ (6,1) &8 &2 &7\\ (0,1,2) &8 &3 &3\\ (1,2,1) &8 &3 &4\\ (2,0,2) &8 &3 &4\\ (3,1,1) &8 &3 &5\\ (5,0,1) &8 &3 &6\\ (0,0,0,2) &8 &4 &2\\ (0,2,0,1) &8 &4 &3\\ (1,0,1,1) &8 &4 &3\\ (2,1,0,1) &8 &4 &4\\ (4,0,0,1) &8 &4 &5\\ (0,0,1,0,1) &8 &5 &2\\ (1,1,0,0,1) &8 &5 &3\\ (3,0,0,0,1) &8 &5 &4\\ (0,1,0,0,0,1) &8 &6 &2\\ (2,0,0,0,0,1) &8 &6 &3\\ (1,0,0,0,0,0,1) &8 &7 &2 & y & y\\ (0,0,0,0,0,0,0,1) &8 &8 &1\\ (9) & 9 & 1 & 9 \\ (1,4) & 9 & 2 & 5 \\ (3,3) & 9 & 2 & 6 \\ (5,2) & 9 & 2 & 7 \\ (7,1) & 9 & 2 & 8 \\ (0,0,3) & 9 & 3 & 3 \\ (0,3,1) & 9 & 3 & 4 \\ (1,1,2) & 9 & 3 & 4 \\ (2,2,1) & 9 & 3 & 5 \\ (3,0,2) & 9 & 3 & 5 \\ (4,1,1) & 9 & 3 & 6 \\ (6,0,1) & 9 & 3 & 7 \\ (0,1,1,1) & 9 & 4 & 3 \\ (1,0,0,2) & 9 & 4 & 3 \\ (1,2,0,1) & 9 & 4 & 4 & y & n \\ (2,0,1,1) & 9 & 4 & 4 \\ (3,1,0,1) & 9 & 4 & 5 \\ (5,0,0,1) & 9 & 4 & 6 \\ (0,0,0,1,1) & 9 & 5 & 2 \\ (0,2,0,0,1) & 9 & 5 & 3 \\ (1,0,1,0,1) & 9 & 5 & 3 \\ (2,1,0,0,1) & 9 & 5 & 4 \\ (4,0,0,0,1) & 9 & 5 & 5 \\ (0,0,1,0,0,1) & 9 & 6 & 2 \\ (1,1,0,0,0,1) & 9 & 6 & 3 \\ (3,0,0,0,0,1) & 9 & 6 & 4 \\ (0,1,0,0,0,0,1) & 9 & 7 & 2 \\ (2,0,0,0,0,0,1) & 9 & 7 & 3 \\ (1,0,0,0,0,0,0,1) & 9 & 8 & 2 \\ (0,0,0,0,0,0,0,0,1) & 9 & 9 & 1 \\ (1,1,1,0,0,1) & 12 & 6 & 4 & y & n \\ (1,3,0,2,0,0,0,1) & 23 & 8 & 7 & y & n\\ we can see that there are likely p(n) aggregates of magnitude n, where p(n) is the nth partition number, this makes sense from the definition of magnitude, this table gives us a convenient way of ranking the entries. Question, can two different entries have exactly the same magnitude, span and fill. Answer, no. If one aggregate has a set of digits, to conserve fill, exactly the same number of digits must be used, but for a different configuration one or more must be moved, thus changing the magnitude. We see that an integer prime p will always have a magnitude p + 1 a span p and a fill 2. We say an integer prime p, as we may be able to extend the notion of primality to the fictional integers above, we can declare any aggregate with fill 2 a prime aggregate, and check their density. We can then tabulate our ’fictional primes’ (2)\\ (1,1)\\ (0,2)\\ (1,0,1)\\ (0,1,1)\\ (1,0,0,1)\\ (0,0,2)\\ (0,1,0,1)\\ (1,0,0,0,1)\\ (0,0,1,1)\\ (0,1,0,0,1)\\ (1,0,0,0,0,1) we can then realise a few (aggregate) products of the above list (1,1)\cdot(0,1,0,1)=(1,2,0,1)=4\\ (0,1,1)\cdot(1,0,0,0,0,1)=(1,1,1,0,0,1)=6\\ (0,0,2)\cdot(1,0,0,0,0,0,0,0,1)=9\\ (0,2)\cdot(0,1,0,1)\cdot(1,0,0,0,0,0,0,1)=8\\ (0,0,1,0,1)\cdot(1,\cdots,1)=15 we then can categorize the natural numbers, how many prime aggregates are needed to construct the numbern, P(n), how many different ways can the number be written in prime aggregates, w(n). For example 4=(1,1)\cdot(0,1,0,1)=(0,2)\cdot(1,0,0,1) so w(4)=2. This means w(1) is zero, w(2)=1, w(3)=1, w(5)=1. In fact any prime p, w(p)=1, we have 6=(0,1,1)\cdot(1,0,0,0,0,1)=(1,1)\cdot(0,0,1,0,0,1)=(1,0,1)\cdot(0,1,0,0,0,1) so w(6)=3. Some questions, are all natural numbers n > 1 able to be created by a finite number of prime aggregates? If the fill of the aggregate representation of n is even then this is true for all n. Is the set of fictional numbers creatable by prime aggregates smaller than the set fictional integers? This is true, as we can create fictional integers with non-even fills. INFERRED RULES If 4 = (1, 2, 0, 1), and 2 = (1, 1), then (0, 1, 0, 1)=2*, a special version of two. However, if 6 = (1, 1, 1, 0, 0, 1) and 3 = (1, 0, 1), then 2* = (0, 1, 0, 0, 0, 1) which is not consistent. Really we should write 4/2 = (0, 1, 0, 1), and 6/3 = (0, 1, 0, 0, 0, 1) where / is aggregate division. These are both prime aggregates. We can then write 8/(4/2)=(1, 2, 0, 0, 0, 0, 0, 1), and (8/(4/2))/2 = (0, 1, 0, 0, 0, 0, 0, 1). In some sense then, these prime aggregates can be named based on the integers. One naming concept is the shifting of prime aggregates, for example 2=(1,1)\\ 2>>1=(0,1,1)\\ This can provide names for stranger aggregates 4/((6/2)<<2)=(0,2) and we gain amazing strings of formulae like 4=2\cdot(3>>1)\\ 6=3\cdot(5>>1)\\ 8=4\cdot(7>>1)\\ 10=5\cdot(9>>1) implying that 2m = m((2m − 1)> > 1). However... 6\cdot(11>>1)=(1,1,1,0,0,1)\cdot(0,1,0,0,0,0,0,0,0,0,0,1)=(1,2,1,0,0,1,0,0,0,0,0,1)\ne12 we are missing a factor of (0, 0, 0, 1)! and we would require 6 ⋅ ((1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1)> > 1) to keep the trend running. This brings us to the observation that 12 cannot be made by a finite number of prime aggregates. \;12=\;(1,2,1,1,0,1,0,0,0,0,0,1)=7 not even. We then see that 14=7\cdot(13>>1)\\ but that 16 is not constructable. STANDARD PRODUCT So far we have missed out the standard numeric product between these aggregates (1,1)\times(1,1)=(1,2,0,1)\\ (1,1)\times(1,0,1)=(1,1,1,0,0,1) which is clearly done by the standard method of pairing each term and adding to the relevant box in the new aggregate. That is C_k= A_iB_j we can then extend this form of product to our prime aggregates above, (1,1)\times(0,2)=(0,1,0,1)\cdot(0,1,0,1)\\ (0,1,1)\times(1,0,0,0,0,1)=(0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1)\\ (1,1)\times(1,0,0,1)=8/(0,2) This product is consistent with multiplication of integers, we would also have the concept of addition from normal integers which makes less sense (1,1)+(1)=(1,0,1)\\ (1,0,1)+(1)=(1,2,0,1)\\ (1,1)+(1,1)=(1,2,0,1)\\ (1,1)+(1,0,1)=(1,0,0,0,1)

© 2016-2024, Kevin J. Black. This work is licensed under a Creative Commons Attribution 4.0 International License.

Let \Pi_N(a,x)= ^N a+x^{p_k} Where pk is the kth prime. Then (\log\Pi_N(a,x) - \log a^N) = {2a}+{3a} -{4a^2} + {5a} + {6a^3} +{7a} -{8a^4}+{9a^3}+}{10a^5}+\cdots It would appear there are a few patterns here. - The power of a in the denominator is the largest proper divisor of the power of x. - The denominator is the power of x. - For prime powers of x the numerator is also the power of x. - For a perfect square or cube power of x, the prime divisor is the numerator. - For a composite power of x the numerator is a polynomial in a, with coefficients of the divisors of x. - After removing the denominator coefficient of the power of x, all raw coefficients present in the above expansion are prime. Focus on the composites x^4 \to -{a^2}\\ x^6 \to {a^3}\\ x^9 \to {a^3}\\ x^{10} \to {a^5}\\ x^{12} \to -{a^6}\\ x^{42} \to }{a^{21}}\\ x^{44} \to -}{a^{22}}\\ x^{30} \to -{a^{15}}\\ x^{66} \to -+11a^{27}}{a^{33}}\\ x^{69} \to }{a^{23}} It seems that for numbers with 3 prime factors, the coefficient of the largest power of a in the numerator is the power of the second largest power of a. Although only a few numbers have been checked so far. It would appear that for a polynomial of the form x^q \to {a^o}\\ xo=q\\ y(o-m)=q\\ z(o-n)=q giving x^q \to {a^{D_m(q)}} (-1)^?pa^{D_m(q)-{p}} where Dm(q) is the largest proper divisor of q. This gives (\log\Pi_N(a,x) - \log a^N) = ^\infty {qa^{D_m(q)}} (-1)^?pa^{D_m(q)-{p}}x^q \\ ^\infty 1+x^{p_k}=\exp\left(^\infty {q} (-1)^?px^q\right) where the pattern for the negative signs is yet to be found. It would appear that if the power of x is divisible by 2 and the power of x divided by 2 is not equal to 1, then the overall sign of the temr in the expansion is negative. Then the polynomials in the numerator must be investigated. It would appear that the polynomial coefficient of 2 is negative if the power of x is of the form 4n + 2, n = 0, 1, 2, 3, .... All other prime divisor polynomial coefficients appear to be positive. This then explains all terms (as far as we know). This gives (\log\Pi_N(a,x) - \log a^N) = ^\infty {qa^{D_m(q)}} (-1)^{q+1}p^*(q)a^{D_m(q)-{p}}x^q \\ ^\infty 1+x^{p_k}=\exp\left(^\infty {q} (-1)^{q+1}p^*(q)x^q\right) where p*(q) means p^*(q)=(-1)^{q/2}\cdot 2,\; p=2\\ p^*(q)=p,\; We may now manipulate this expression ^\infty 1+x^{p_k}=\exp\left(\left[}^\infty {q} (-1)^{q+1}p^*(q)x^q\right] + \left[}^\infty {q} (-1)^{q+1}p^*(q)x^q\right]\right) \\ ^\infty 1+x^{p_k}=\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\\ ^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}=\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\\ \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)=}^\infty {q} (-1)^{q+1}p^*(q)x^q\\ \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)=}^\infty x^q Where the right hand side is now the ordinary generating function for primes. Then of course we also have \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty x^q\right)}\right)=}^\infty {q} (-1)^{q+1}p^*(q)x^q Then if we set x = 1, and change the infinite sums to n (checked that the series are generated at the same rate) we should have have the prime counting function \pi(n) = \left.\log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)\right|_{x=1}

In this paper we propose a tokenization algorithm of Reversible Hybrid type, as defined in PCI DSS guidelines for designing a tokenization solution, based on a block cipher with a secret key and (possibly public) additional input. We provide some formal proofs of security for it, which imply our algorithm satisfies the most significant security requirements described in PCI DSS tokenization guidelines. Finally, we give an instantiation with concrete cryptographic primitives and fixed length of the PAN, and we analyze its efficiency and security.

WHAT IS LATEX? LaTeX is a programming language that can be used for writing and typesetting documents. It is especially useful to write mathematical notation such as equations and formulae. HOW TO USE LATEX TO WRITE MATHEMATICAL NOTATION There are three ways to enter “math mode” and present a mathematical expression in LaTeX: 1. _inline_ (in the middle of a text line) 2. as an _equation_, on a separate dedicated line 3. as a full-sized inline expression (_displaystyle_) _inline_ Inline expressions occur in the middle of a sentence. To produce an inline expression, place the math expression between dollar signs ($). For example, typing $E=mc^2$ yields E = mc². _equation_ Equations are mathematical expressions that are given their own line and are centered on the page. These are usually used for important equations that deserve to be showcased on their own line or for large equations that cannot fit inline. To produce an inline expression, place the mathematical expression between the symbols \[! and \verb!\]. Typing \[x=}{2a}\] yields \[x=}{2a}\] _displaystyle_ To get full-sized inline mathematical expressions use \displaystyle. Typing I want this $\displaystyle ^{\infty} {n}$, not this $^{\infty} {n}$. yields: I want this $\displaystyle ^{\infty}{n}$, not this $^{\infty}{n}.$ SYMBOLS (IN _MATH_ MODE) The basics As discussed above math mode in LaTeX happens inside the dollar signs ($...$), inside the square brackets \[...\] and inside equation and displaystyle environments. Here’s a cheatsheet showing what is possible in a math environment: -------------------------- ----------------- --------------- _description_ _command_ _output_ addition + + subtraction - − plus or minus \pm ± multiplication (times) \times × multiplication (dot) \cdot ⋅ division symbol \div ÷ division (slash) / / simple text text infinity \infty ∞ dots 1,2,3,\ldots 1, 2, 3, … dots 1+2+3+\cdots 1 + 2 + 3 + ⋯ fraction {b} ${b}$ square root $$ nth root \sqrt[n]{x} $\sqrt[n]{x}$ exponentiation a^b ab subscript a_b ab absolute value |x| |x| natural log \ln(x) ln(x) logarithms b logab exponential function e^x=\exp(x) ex = exp(x) deg \deg(f) deg(f) degree \degree $\degree$ arcmin ^\prime ′ arcsec ^{\prime\prime} ′′ circle plus \oplus ⊕ circle times \otimes ⊗ equal = = not equal \ne ≠ less than < < less than or equal to \le ≤ greater than or equal to \ge ≥ approximately equal to \approx ≈ -------------------------- ----------------- ---------------

Electrophysiology is entering the era of ‘Big Data’. Multiple probes, each with hundreds to thousands of individual electrodes, are now capable of simultaneously recording from many brain regions. The major challenge confronting these new technologies is transforming the raw data into physiologically meaningful signals, i.e. single unit spikes. Sorting the spike events of individual neurons from a spatiotemporally dense sampling of the extracellular electric field is a problem that has attracted much attention , but is still far from solved. Current methods still rely on human input and thus become unfeasible as the size of the data sets grow exponentially. Here we introduce the t-student stochastic neighbor embedding (t-sne) dimensionality reduction method as a visualization tool in the spike sorting process. T-sne embeds the n-dimensional extracellular spikes (n = number of features by which each spike is decomposed) into a low (usually two) dimensional space. We show that such embeddings, even starting from different feature spaces, form obvious clusters of spikes that can be easily visualized and manually delineated with a high degree of precision. We propose that these clusters represent single units and test this assertion by applying our algorithm on labeled data sets both from hybrid and paired juxtacellular/extracellular recordings . We have released a graphical user interface (gui) written in python as a tool for the manual clustering of the t-sne embedded spikes and as a tool for an informed overview and fast manual curation of results from other clustering algorithms. Furthermore, the generated visualizations offer evidence in favor of the use of probes with higher density and smaller electrodes. They also graphically demonstrate the diverse nature of the sorting problem when spikes are recorded with different methods and arise from regions with different background spiking statistics.

Predicting the binding affinity between MHC proteins and their peptide ligands is a key problem in computational immunology. State of the art performance is currently achieved by the allele-specific predictor NetMHC and the pan-allele predictor NetMHCpan, both of which are ensembles of shallow neural networks. We explore an intermediate between allele-specific and pan-allele prediction: training allele-specific predictors with synthetic samples generated by imputation of the peptide-MHC affinity matrix. We find that the imputation strategy is useful on alleles with very little training data. We have implemented our predictor as an open-source software package called MHCflurry and show that MHCflurry achieves competitive performance to NetMHC and NetMHCpan.

How many people use the typesetting language LaTeX? This is obviously a hard question. However, another way to look at it is to calculate the percentage of published scholarly articles written in LaTeX.

ABSTRACT A p-rough number for some prime p, is a member of the set of numbers not divisible by any prime q < p. This document will note down the generating functions for the first few prime-rough numbers, and discuss their inverse Z-transforms, i.e a function a(n) that gives the nth member of the set. INTRODUCTION The 3-rough numbers are simply the odd numbers 1, 3, 5, 7, 9, 11⋯, the numbers which are not divisible by p ∈ {2}, See the table below for the appropriate OEIS label. {|c|c|} \hline p & p \\ \hline 2 & A000027 & 1,2,3,4,5,6,7\\ 3 & A005408 & 1,3,5,7,9,11,13\\ 5 & A007310 & 1,5,7,11,13,17,19\\ 7 & A007775 & 1, 7, 11, 13, 17, 19, 23\\ 11 & A008364& 1, 11, 13, 17, 19, 23, 29\\ 13 & A008365& 1, 13, 17, 19, 23, 29, 31\\ 17 & A008366& 1, 17, 19, 23, 29, 31, 37,\\ 19 & A166061& 1, 19, 23, 29, 31, 37, 41\\ 23 & A166063& 1, 23, 29, 31, 37, 41, 43\\ \hline From the table above we can see the first entry is always 1, the second entry is the nth prime, (i.e the index in the name of that sequence) and the next entry is the name of the next sequence. There are an increasing number of primes in each sequence, rather the probability of finding a prime number is higher. GENERATING FUNCTIONS A generating function for a sequence with terms a(n) is simply the series, G_a(z)=a(0)z^0 + a(1)z^1 + a(2)z^2 + \cdots= ^\infty a(n)z^n, however, for some sequences it makes more sense to count from 1, giving the description G_a(z)=a(1)z^1 + a(2)z^2 + \cdots= ^\infty a(n)z^n, the latter is the description we will use. We can see that the generating function for the natural numbers 1, 2, 3, ⋯ (which are also the 2-rough numbers) is then G_2(z)={(z-1)^2}=z+2z^2+3z^3+\cdots\\ a_2(n)=n Then for the 3-rough numbers we have the odd numbers, G_3(z)={(z-1)^2} \\ a_3(n) = 2n-1 we can use the Mathematica expression \\ to extract the sequence function a₃(n). For the 5-rough numbers we have numbers congruent to 1 or 5 mod 6, G_5(z) ={(1-z)^2(1+z)}\\ a_5(n) = {2}(6n+(-1)^n-3) From these we notice that G_3(z)=G_2(z)(1+z)\\ G_5(z)={G_3(z)}(1+4z+z^2) From here on things get less simple. The next sequence is the 7-rough numbers G_7(z)={(1+z)(z^2+1)(z^4+1)(z-1)^2} But the sequence description a(n) is terribly complicated a_7(n)= {8}e^{-{4}in\pi}\left((1-3i)+(2+i)+(-1)^n\left((1-3i)-(2+i)+((i-14)+30n)\cos(n\pi/4)+(2+6i)\cos(n\pi/2)+(2-i)\cos(3n\pi/4)+((16i-1)-30in)\sin(n\pi/4)+(2+4i)\sin(n\pi/2)-\sin(3n\pi/4)\right)\right) Then we have the 11-rough numbers G_{11}(z) = + 10z^{47} + 2z^{46} + 4z^{45} + 2z^{44} + 4z^{43} + 6z^{42} + 2z^{41} + 6z^{40} + 4z^{39} + 2z^{38} + 4z^{37} + 6z^{36} + 6z^{35} + 2z^{34} + 6z^{33} + 4z^{32} + 2z^{31} + 6z^{30} + 4z^{29} + 6z^{28} + 8z^{27} + 4z^{26} + 2z^{25} + 4z^{24} + 2z^{23} + 4z^{22} + 8z^{21} + 6z^{20} + 4z^{19} + 6z^{18} + 2z^{17} + 4z^{16} + 6z^{15} + 2z^{14} + 6z^{13} + 6z^{12} + 4z^{11} + 2z^{10} + 4z^9 + 6z^8 + 2z^7 + 6z^6 + 4z^5 + 2z^4 + 4z^3 + 2z^2 + 10z + 1)}{(z^{49} - z^{48} - z + 1)} \\ G_{11}(z) = +1 + 2(z^{46}+ z^{44}+ z^{41}+ z^{38}+ z^{34}+ z^{31}+ z^{25}+ z^{23}+ z^{17}+ z^{14}+ z^{10}+ z^7+ z^4+ z^2) + 4(z^{45} + z^{43} + z^{39} + z^{37} + z^{32}+ z^{29}+ z^{26} + z^{24} + z^{22} + z^{19} + z^{16} + z^{11}+z^9+z^5+z^3) + 6(z^{42} + z^{40} + z^{36} + z^{35} + z^{33} + z^{30} + z^{28} + z^{20} + z^{18} + z^{15} + z^{13} + z^{12} + z^8 + z^6) + 8(z^{27} + z^{21})+ 10(z^{47}+z) )}{(z^{49} - z^{48} - z + 1)} Whose sequence description takes us a few screens and is silly to place here... For this we have learnt that the generating functions remain somewhat simple compared to the sequence descriptions.

Cosmological parameter estimation is traditionally performed in the Bayesian context. By adopting an “agnostic” statistical point of view, we show the interest of confronting the Bayesian results to a frequentist approach based on profile-likelihoods. To this purpose, we have developed the _Cosmological Analysis with a Minuit Exploration of the Likelihood_ () software. Written from scratch in pure C++, emphasis was put in building a clean and carefully-designed project where new data and/or cosmological computations can be easily included. CAMEL incorporates the latest cosmological likelihoods and gives access _from the very same input file_ to several estimation methods: - A high quality Maximum Likelihood Estimate (a.k.a “best fit”) using , - profile likelihoods, - a new implementation of an Adaptive Metropolis MCMC algorithm that relieves the burden of reconstructing the proposal distribution. We present here those various statistical techniques and roll out a full use-case that can then used as a tutorial. We revisit the  parameters determination with the latest  data and give results with both methodologies. Furthermore, by comparing the Bayesian and frequentist approaches, we discuss a “likelihood volume effect” that affects the optical reionization depth when analyzing the high multipoles part of the  data. The software, used in several  data analyzes, is available from http://camel.in2p3.fr. Using it does not require advanced C++ skills.

We compare the properties of giant molecular associations in the galaxy Messier 100 (M100) with those of the less massive giant molecular clouds in the Milky Way and Local Group, while also observing how those properties change within M100 itself. From this analysis of cloud mass, radius, and velocity dispersion, we determine that the clouds are in or near virial equilibrium and that their properties are consistent with the underlying trends for the Milky Way. We find differences between nuclear, arm and inter-arm M100 populations, such as the nuclear clouds being the most massive and turbulent, and arm and inter-arm populations having differently shaped mass distributions from one another. Through the analysis of velocity gradients, cloud motion can be attributed to turbulence rather than large scale shearing motion. This is supported by our comparison with turbulence regulated star formation models. Finally, we calculate ISM depletion times to see how quickly clouds turn gas into stars and found that clouds form stars more efficiently if they are turbulent or dense.

A lot of attention has been given to the consequences of the latest strong El Niño event. People often talk about meteorological phenomena as El Niño (or La Niña) conditions, but what are these, and how do we come about our notions of what is a ’typical’ El Niño event? How consistent do we expect the effects of this phenomena to be, especially when these ’signature effects’ occur thousands of kilometers away from the Pacific Ocean? Often understanding about the typical effects of large scale climate variations are derived from _composites_. This is a common statistical method where elements are classified into groups based on some external consideration, and then the properties of each group is expressed by the average of all the elements it contains. This can be a very efficient way to visualize large data sets, but it can also imply more consistency within groups than is actually the case. This post goes over some of mechanics of creating composites, and ways to explore to what degree they can be taken at ’face value’.

IN A NUTSHELL: Gravitational waves are ripples in the fabric of space time produced by violent events, like merging together two black holes or the explosion of a massive star. Unlike light (electromagnetic waves) gravitational waves are not absorbed or altered by intervening material, so they are very clean proxies of the physical process that produced them. They are expected to travel at the speed of light and, if detected, they could give precious information about the cataclysmic processes that originated them and the very nature of gravity. That’s why the direct detection of gravitational waves is such an important endeavor. Definitely worthy of a Nobel prize in physics.

This article presents highlights chosen from research that appeared during 2016 on Tourette syndrome and other tic disorders. Selected articles felt to represent meaningful advances in the field are briefly summarized.

This article describes the generation of elastic shear waves in a soft medium using a laser beam. Our experiments show two different regimes depending on laser energy. Physical modeling of the underlying phenomena reveals a thermoelastic regime caused by a local dilatation resulting from temperature increase, and an ablative regime caused by a partial vaporization of the medium by the laser. Computed theoretical displacements are close to experimental measurements. A numerical study based on the physical modeling gives propagation patterns comparable to those generated experimentally. These results provide a physical basis for the feasibility of a shear wave elastography technique (a technique which measures a soft solid stiffness from shear wave propagation) by using a laser beam.

Pharmit (http://pharmit.csb.pitt.edu) provides an online, interactive environment for the virtual screening of large compound databases using pharmacophores, molecular shape, and energy minimization. Users can import, create, and edit virtual screening queries in an interactive browser-based interface. Queries are specified in terms of a pharmacophore, a spatial arrangement of the essential features of an interaction, and molecular shape. Search results can be further ranked and filtered using energy minimization. In addition to a number of pre-built databases of popular compound libraries, users may submit their own compound libraries for screening. Pharmit uses state-of-the-art sub-linear algorithms to provide interactive screening of millions of compounds. Queries typically take a few seconds to a few minutes depending on their complexity. This allows users to iteratively refine their search during a single session. The easy access to large chemical datasets provided by Pharmit simplifies and accelerates structure-based drug design. Pharmit is available under a dual BSD/GPL open-source license.

ABSTRACT A certain conjecture has come to my attention which I have constructed purely from observation. It uses the syntax a; b is a repeated b times, i.e a; 4 = aaaa, this holds for larger strings, abc; 3 = abcabcabc. where a, b, c can be replace by any ’digit’ in any reasonable integer base. CONJECTURE I conjecture that \;{b}\in then {b}\in with the condition k \;\; (b-1) = a where s is any string of numbers, a is an integer and b is a prime.

ENERGY CONSERVATION IN SWEPT FIELD LIMIT For a calorimeter sample (plus addenda) weakly thermally linked to a temperature controlled reservoir, energy conservation implies -T dS = \kappa \Delta T dt + C_{} dT where κ is the sample to reservoir thermal conductance and addenda is the heat capacity of the actual addenda (such as the thermometer, heater, and glue or grease binding the sample to the sensors) plus the heat capacity of the sample lattice (due to phonons). The left hand side term is the heat released by the system — which in the case of a spin system, for example, would be the heat released by the spins — when the field is changed by dH. The minus sign indicates that system entropy decreases as heat is released. Most of the released heat flows to the reservoir but some fraction heats up the addenda (to the same temperature as the system). The first term on the right hand side describes heat flow to the reservoir. The second term describes the temperature rise of the addenda. In a non-adiabatic relaxation-time or ac-calorimeter like that used in our swept-field measurements , the first term dominates. In contrast, in an adiabatic measurement, the first term is negligible.

East Africa has two rainy seasons: the long rains (March–May, MAM) and the short rains (October–December, OND). Most CMIP3/5 coupled models overestimate the short rains while underestimating the long rains. In this study, the East African rainfall bias is investigated by comparing the coupled historical simulations from CMIP5 to the corresponding SST-forced AMIP simulations. Much of the investigation is focused on the MRI-CGCM3 model, which successfully reproduces the observed rainfall annual cycle in East Africa in the AMIP experiment but its coupled historical simulation has a similar but stronger bias as the coupled multimodel mean. The historical−AMIP monthly climatology rainfall bias in East Africa can be explained by the bias in the convective instability (CI), which is dominated by the near surface moisture static energy (MSE) and ultimately by the MSE’s moisture component. The near surface MSE bias is modulated by the sea surface temperature (SST) over the western Indian Ocean. The warm SST bias in OND can be explained by both insufficient ocean dynamical cooling and latent flux, while the insufficient short wave radiation and excess latent heat flux mainly contribute to the cool SST bias in MAM.

ABSTRACT Knowledge of genetic cause in neurodevelopmental disorders can highlight molecular and cellular processes critical for typical development. Furthermore, the relative homogeneity of neurodevelopmental disorders of known genetic origin allows the researcher to establish the subsequent neurobiological processes that mediate cognitive and behavioural outcomes. The current study investigated white matter structural connectivity in a group of individuals with intellectual disability due to mutations in _ZDHHC9_. In addition to shared cause of cognitive impairment, these individuals have a shared cognitive profile, involving oro-motor control difficulties and expressive language impairment. Analysis of structural network properties using graph theory measures showed global reductions in mean clustering coefficient and efficiency in the _ZDHHC9_ group, with maximal differences in frontal and parietal areas. Regional variation in clustering coefficient across cortical regions in _ZDHHC9_ mutation cases was significantly associated with known pattern of expression of _ZDHHC9_ in the normal adult human brain. The results demonstrate that a mutation in a single gene impacts upon white matter organisation across the whole-brain, but also shows regionally specific effects, according to variation in gene expression. Furthermore, these regionally specific patterns may link to specific developmental mechanisms, and correspond to specific cognitive deficits.

Summary Both classical taxonomy and DNA barcoding are engaged in the task of digitising the living world. Much of the taxonomic literature remains undigitised. The rise of open access publishing this century, and the freeing of older literature from the shackles of copyright has greatly increased the online availability of taxonomic descriptions, but much of the literature of the mid- to late twentieth century remains offline ("dark texts"). DNA barcoding is generating a wealth of computable data that in many ways is much easier to work with than classical taxonomic descriptions, but many of the sequences are not identified to species level. These "dark taxa" hamper the classical method of integrating biodiversity data using shared taxonomic names. Voucher specimens are a potential common currency of both the taxonomic literature and sequence databases, and could be used to help link names, literature, and sequences. An obstacle to this approach is the lack of stable, resolvable specimen identifiers. The paper concludes with an appeal for a global "digital dashboard" to assess the extent to which biodiversity data is available online. Keywords: DNA barcoding, taxonomy, dark taxa, dark texts, digitisation