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bibkey: hanna2026a393867 authors: Paul D. Hanna year: 2026 title: “OEIS A393867, logarithmic derivative of A393866 (g.f. with [x^n] A^prime(n) = prime(n) [x^(n-1)] A^prime(n))” doi: null url: https://oeis.org/A393867 claim: “G.f. A(x) satisfies [x^n] A(x)^prime(n) = prime(n) * [x^(n-1)] A(x)^prime(n) for n >= 1. Logarithmic derivative of A393866. Conjecture: a(n) is divisible by prime(n) for n > 1. Conjecture: all terms are odd.” strata_touched:

  • D5/S1/Recurrence/Parity/PrimePowerShiftLogDerivative
  • D5/S1/Recurrence/Parity/PrimePowerShiftLogDerivativeOdd license: citation-only triage: anchor

OEIS A393867

Paul D. Hanna’s entries A393866 and A393867, dated March 29, 2026, specify the coefficient equations and the divisibility conjecture quoted above. Here prime(n) is the nth prime, with prime(1) = 2. The series F has constant coefficient one and integer coefficients. The logarithmic derivative sequence is indexed by a(n) = [x^(n-1)] F'(x)/F(x) for n >= 1.

The target is A393867’s divisibility comment. Formula (2) of A393866 instead prints [x^n] A'(x)/A(x) == 0 (mod prime(n)) for n > 1; its coefficient index differs from the divisibility target.

The formal module constructs the unique integer solution to the defining equations and proves A393867’s divisibility conjecture. Binomial divisibility makes each recursive correction integral, and the formal derivative power rule isolates the logarithmic-derivative coefficient modulo the nth prime. The companion printed_formula_false refutes A393866’s printed formula at n = 2: the coefficient of degree two in F'/F is 25, whereas prime(2) = 3.

The companion module PrimePowerShiftLogDerivativeOdd addresses exactly the second comment, “Conjecture: all terms are odd.” It proves consecutive even-odd coefficients of F agree modulo two, then deduces (1+X)F'=F and (1+X)(F'/F)=1 over ZMod 2. This is a repository proof, not an attributed proof from OEIS. Direct OEIS JSON retrieval on 2026-09-10 verified both NAME lines, both conjecture comments, and the DATA prefixes. The explicit search scope and zero-result arXiv/GitHub queries are recorded in Problems/oeis-a393867-all-terms-odd.md; no priority claim is made.

Verified locator

  • URL: https://oeis.org/A393867
  • Defining series: https://oeis.org/A393866