A328538 Second members of a list of seven pairs connected with parity of sums of partition numbers in arithmetic progressions.

8, 12, 24, 40, 48, 120, 168

Offset

1,1

Comments

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

"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.

Here p(n) is the number of partitions of n, A000041.

References

Ballantine, Cristina, and Mircea Merca. "Parity of sums of partition numbers and squares in arithmetic progressions." The Ramanujan Journal 44.3 (2017): 617-630.

Links

Table of n, a(n) for n=1..7.

Letong Hong and Shengtong Zhang, Proof of the Ballantine-Merca Conjecture and theta function identities modulo 2, arXiv:2101.09846 [math.NT], 2021.

Crossrefs

Cf. A000041, A328537.

Sequence in context: A088525 A054735 A162691 * A077566 A241482 A368124

Adjacent sequences: A328535 A328536 A328537 * A328539 A328540 A328541

Keyword

nonn,fini,full

Author

N. J. A. Sloane, Oct 18 2019

Status

approved