A Proof of Bala's Congruence Conjectures for A158690
Abstract
Let a(n) be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function ∑n≥0 a(n)tn/n! = 1+∑m≥1∏j=1m(1−e−(2j−1)t). We prove two congruence conjectures of Peter Bala. The first states that, for every integer k≥1, the sequence a(n) modulo k is eventually periodic with period dividing φ(k). We prove the stronger statement that the Carmichael function λ(k) is an eventual period. The second conjecture asserts the shifted Gauss congruences a(npr+i)≡ a(npr−1+i)pmodpr for every i≥0, every prime p, and all n,r≥1. Both results follow from a general theorem for exponential generating functions of the form G(et−1) with G∈ℤ[[y]], together with the standard power-sum formula for Stirling numbers of the second kind.
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BibTeX
@misc{kallat2026proof,
title = {{A Proof of Bala's Congruence Conjectures for A158690}},
author = {Ahaan Kallat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07238},
eprint = {2609.07238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07238},
}
RIS
TY - UNPB
TI - A Proof of Bala's Congruence Conjectures for A158690
AU - Ahaan Kallat
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07238
UR - https://arxiv.org/abs/2609.07238
AB - Let a(n) be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function ∑_n≥0 a(n)t^n/n! = 1+∑_m≥1∏_j=1^m(1−e^−(2j−1)t). We prove two congruence conjectures of Peter Bala. The first states that, for every integer k≥1, the sequence a(n) modulo k is eventually periodic with period dividing φ(k). We prove the stronger statement that the Carmichael function λ(k) is an eventual period. The second conjecture asserts the shifted Gauss congruences a(np^r+i)≡ a(np^r−1+i)pmodp^r for every i≥0, every prime p, and all n,r≥1. Both results follow from a general theorem for exponential generating functions of the form G(e^t−1) with G∈ℤ[[y]], together with the standard power-sum formula for Stirling numbers of the second kind.
ER -