A328537 First members of a list of seven pairs connected with parity of sums of partition numbers in arithmetic progressions.

6, 8, 12, 15, 16, 20, 21

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, A328538.

Sequence in context: A315860 A315861 A027827 * A157941 A059611 A177085

Adjacent sequences: A328534 A328535 A328536 * A328538 A328539 A328540

Keyword

nonn,fini,full

Author

N. J. A. Sloane, Oct 18 2019

Status

approved