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A302056 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^4 is zero.

%I A302056 #16 Feb 12 2021 17:51:32
%S A302056 9,14,19,24,31,34,39,42,44,49,53,59,64,65,69,74,75,82,84,86,89,94,97,
%T A302056 99,108,109,111,114,116,119,124,130,133,134,139,144,149,150,152,157,
%U A302056 159,163,164,167,169,174,180,184,185,189,194,196,198,199,201,203,207,209
%N A302056 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^4 is zero.
%C A302056 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.
%C A302056 From _Jianing Song_, Feb 09 2021: (Start)
%C A302056 The following are equivalent:
%C A302056 - k is in this sequence;
%C A302056 - At least one prime congruent to 5 modulo 6 divides 6*k+1 with an odd exponent;
%C A302056 - 6*k+1 is not of the form x^2 + x*y + y^2, i.e., 6*k+1 is in A034020. (End)
%H A302056 Seiichi Manyama, <a href="/A302056/b302056.txt">Table of n, a(n) for n = 1..10000</a>
%H A302056 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%t A302056 Flatten[Position[nmax = 210; Rest[CoefficientList[Series[QPochhammer[x]^4, {x, 0, nmax}], x]], 0]]
%t A302056 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]]
%t A302056 Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Exp[-4 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
%o A302056 (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
%Y A302056 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m = 1), A213250 (m = 2), A014132 (m = 3), this sequence (m = 4), A302057 (m = 5), A020757 (m = 6), A322430 (m = 8), A322431 (m = 10), A322432 (m = 14), A322043 (m = 15), A322433 (m = 26).
%Y A302056 Cf. A000727, A034020.
%K A302056 nonn
%O A302056 1,1
%A A302056 _Ilya Gutkovskiy_, Mar 31 2018