Metadata
bibkey: hanna2025a382319
authors: Paul D. Hanna
year: 2025
title: "OEIS A382319, g.f. A(x) satisfies A(x) = x/(1-x) + Sum_{n>=1} A(x^n)^3"
doi: null
url: https://oeis.org/A382319
claim: "Conjecture: a(n) == A382318(3*n) (mod 3) for n >= 0."
strata_touched:
- D5/S1/Recurrence/Invariants/HannaLambertCubeModThree
- D5/S1/Recurrence/Invariants/HannaLambertCubeModThreeConverse
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A382319
Source statement
P. D. Hanna, OEIS A382319, Name (unpaginated):
G.f. A(x) satisfies A(x) = x/(1-x) + Sum_{n>=1} A(x^n)^3.
Comments (unpaginated):
Conjecture: a(n) == A382318(3*n) (mod 3) for n >= 0.
Both entries have offset one. The rational power series have zero constant coefficient. The infinite sum is interpreted coefficientwise: at degree m, sum coeff(m/n,F) over the positive divisors n of m; at m=0 the finite sum is empty. In A382319 the cube is taken before Lambert summation.
Mathematical relation
The two equations imply B = Sum_{n>=1} A(x^n) and A = x+B^3. The integral fixed point is constructed by coefficient stabilization. Cubic Frobenius modulo three identifies B(x)^3 with B(x^3), which gives the coefficient congruences. The rational series and integer residues are the objects stated by the two formal propositions.
Computed evidence
The independent coefficient checker is in
the-omega-institute/trureturing-experiments,
docs/reports/oeis-a382318-a382319-a382124-a382125-hanna-frobenius-mod3/,
commit 7b61b379bca979d9231306df7e0b20924bcc3491 (merge
24caf615e0ee3e77c2e10d7b682c951b239efa1a). Its check.py SHA-256 is
319ee58b884641c875833eb3e830906af72d593ab73e1d6b0da6ff155cd352f3.
Running python3 check.py 300 in that directory exits 0 with RESULT PASS.
This finite computation checks coefficients through degree 300; it is not
an assertion about all coefficients.
Literature boundary
The source is an explicitly conjectural OEIS statement. The literature screen in issue #18862 found no published proof or refutation; completeness and continued external open status are ASSUMED-UNVERIFIED. The prime-exponent extension and asymptotic formula remain outside these formal propositions.