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,\;