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%I #8 Jan 26 2021 05:57:27
%S 8,12,24,40,48,120,168
%N Second members of a list of seven pairs connected with parity of sums of partition numbers in arithmetic progressions.
%C The seven pairs are [6,8], [8,12], [12,24], [15,40], [16,48], [20,120], [21,168]. The list is definite, but the conjecture is unproved. The conjecture asserts that
%C "Sum_{ak+1 square} p(n-k) == 1 mod 2 if and only if bn+1 is a square" holds if and only if [a,b] is one of these seven pairs.
%C Here p(n) is the number of partitions of n, A000041.
%D Ballantine, Cristina, and Mircea Merca. "Parity of sums of partition numbers and squares in arithmetic progressions." The Ramanujan Journal 44.3 (2017): 617-630.
%H Letong Hong and Shengtong Zhang, <a href="https://arxiv.org/abs/2101.09846">Proof of the Ballantine-Merca Conjecture and theta function identities modulo 2</a>, arXiv:2101.09846 [math.NT], 2021.
%Y Cf. A000041, A328537.
%K nonn,fini,full
%O 1,1
%A _N. J. A. Sloane_, Oct 18 2019
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