Metadata
bibkey: tebni2022a350759
authors: Mélika Tebni
year: 2022
title: OEIS A350759, COMMENTS prime congruences
doi: null
url: https://oeis.org/A350759
claim: Both prime congruences for the integer sequence with exponential generating function exp(-1+(1+x)*(1-log(1+x))).
strata_touched:
- D5/S3/Arith/Congruence/TebniA350759PrimeCongruences
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A350759
The source is OEIS A350759, revision 17, COMMENTS and FORMULA, unpaginated. No arXiv identifier or DOI is identified for this source.
Source statements
Conjectures: For all p prime, (a(p) + a(p+1) - 2) == 0 (mod p), a(p+1) == 1 (mod ((p+1)*p)).
E.g.f.: exp(-1 + (1 + x)*(1 - log(1 + x))).
The second COMMENTS line inherits the prime quantifier of the first line.
Indices are natural numbers beginning at zero, and the coefficients are integers.
The formal statement includes existence of an integer sequence with this exact
exponential generating function and quantifies over every such sequence.
Mathlib’s PowerSeries.log ℚ denotes the formal series for log(1+X);
PowerSeries.subst inner outer substitutes into the outer series.
Mathematical scope
The coefficient argument separates the term of expansion index zero from the
remaining terms. Factorial scaling makes the former equal to one and the latter
integer multiples of the coefficient index. The differential equation
F' = -log(1+X) * F, prime binomial divisibility and Wilson’s theorem give the
prime-successor congruence. Coprimality of consecutive integers gives the product
modulus in the second assertion.
Paul D. Hanna’s FORMULA in OEIS A176118 already states the rising-factorial expansion used in this argument:
E.g.f.: Sum_{n>=0} (-1)^n * x^n/n! * Product_{k=1..n} (k + x).
The expansion is credited prior art. The scope here is the two A350759 COMMENTS assertions; composite-index characterizations and higher prime-power refinements remain separate questions. Completeness of the literature screen is ASSUMED-UNVERIFIED.
The OEIS material is cited with attribution; this note does not redistribute the OEIS database or the independently supplied experiment program.