Divisibility by linear forms in two integral factorial ratios: Bala's conjectures for A295431 and A295432
Year: 2026
Type: preprint
Abstract: This paper studies the divisibility of two sequences of integral factorial ratios, OEIS A295431 and A295432, by polynomials of degree one in the index. These are the first two of the fifty-two sporadic integral factorial ratios in Bober’s classification of the ratios of height one whose step function takes only the values zero and one. P. Bala conjectured several divisibility properties of these sequences, some of them with explicit integer multipliers, together with two general statements on the existence of multipliers for products of linear forms. All these conjectures are proved. The proofs rest on a general criterion, derived from Legendre’s formula, which gives an explicit integer multiplier for the quotient of a balanced integral factorial ratio by a product of linear forms, provided that the step function of the ratio is positive on one side of certain rational points. The multiplier is sufficient but need not be the smallest one. For the primes two and three in A295431, closed expressions for the valuations in terms of digit sums are also used. The smallest integer multipliers are determined in the cases where Bala proposed explicit values and for the four linear factors in the recurrence of A295432. For one linear form to which the criterion does not apply, it is shown, by means of primes in arithmetic progressions, that no multiplier exists. A verification program in Python, using exact arithmetic, accompanies the paper. This is the English-language version of the paper. A Spanish-language version is available at DOI: 10.5281/zenodo.24010375
Author: Cerdá Bennassar, Miguel
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225401595