bibkey: hanna2004a091190
authors: Paul D. Hanna
year: 2004
title: "OEIS A091190 and A091200: the Somos prime integrality family"
doi: null
url: https://oeis.org/A091190
claim: "More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime."
strata_touched:
  - D5/S1/Recurrence/Invariants/SomosPrimeIntegrality
license: citation-only
triage: anchor

OEIS A091190 and A091200: the Somos prime integrality family

Verified locator

Source: https://oeis.org/A091190

Statement

More generally, given A(x) satisfies xA(x)^p = B(xA(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime.

The entries are unpaginated. A091190 gives:

G.f. A(x) satisfies xA(x)^3 = B(xA(x^3)) where B(x) = x/(1 - 3*x).

The shared comment continues:

This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) studied by Michael Somos

The normalized rational series has constant coefficient one, as in the offset-zero entries. The source expression for B is encoded coefficientwise by mk fun n => if n = 0 then 0 else (p : ℚ) ^ (n - 1). This is the existing scaled geometric series, whose denominator identity is reused. The claim covers p at least two, rational existence, and the equivalence of coefficientwise integrality with primality. Frobenius gives integral lifting at prime exponents; the low-degree differential recurrence contradicts integrality modulo the least prime factor of a composite exponent. No statement about other choices of B is asserted.

Literature-screen completeness is ASSUMED-UNVERIFIED. The source entries cite no proof of these conjectures; this is a statement about the checked entries, not a claim of exhaustive literature coverage.