OFFSET
1,1
COMMENTS
Numbers k such that number of partitions of k into an even number of distinct parts equals number of partitions of k into an odd number of distinct parts, with 4 types of each part.
From Jianing Song, Feb 09 2021: (Start)
The following are equivalent:
- k is in this sequence;
- At least one prime congruent to 5 modulo 6 divides 6*k+1 with an odd exponent;
- 6*k+1 is not of the form x^2 + x*y + y^2, i.e., 6*k+1 is in A034020. (End)
LINKS
MATHEMATICA
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[QPochhammer[x]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Exp[-4 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
PROG
(PARI) x='x+O('x^999); v=Vec(eta(x)^4 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Altug Alkan, Mar 31 2018, after Joerg Arndt at A213250
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 31 2018
STATUS
approved