bibkey: bala2022egfgeneral authors: Peter Bala year: 2022 title: “Bala’s general totient-period conjecture for integral G(exp(x) - 1)” doi: null url: https://oeis.org/A305550 claim: “More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. The property is eventual periodicity modulo every positive k with period dividing phi(k).” strata_touched:

• D5/S1/Recurrence/Periodic/IntegralEgfTotientPeriod license: citation-only triage: anchor

Bala’s general exponential generating function conjecture

Peter Bala’s July 8, 2022 comment on A305550 contains the broader conjecture quoted above. The same broader conjecture appears on A004123. These are two locations of one conjecture, which this module settles once, not twice.

For any integral coefficient function g, let G be the rational power series with coefficients g(k). The module proves that n! times the coefficient of x^n in G(exp(x) - 1) equals the integer sum of g(k) k! S(n,k) for k at most n. In particular, all these factorial-scaled coefficients are integers; no separate integrality hypothesis is needed. The imported weighted Stirling-transform theorem gives period phi(m) modulo m from n at least m for every positive m. The integer sequence is extracted from the rational coefficients and proved equal to T(g,n); it is not assumed to satisfy an unproved coefficient recurrence.

Verified locator

• URL: https://oeis.org/A305550
• Locator: Peter Bala’s July 8, 2022 COMMENT, beginning “More generally”.
• Duplicate location of the same conjecture: https://oeis.org/A004123