bibkey: hanna2026a396806 authors: Paul D. Hanna year: 2026 title: “OEIS A396806: iterated-exponential congruences” doi: null url: https://oeis.org/A396806 claim: “For A396806 and every n >= 1, a(n) is odd iff n is odd, and a(n) is congruent to n modulo 3 and modulo 6.” strata_touched:
- D5/S1/Recurrence/Residue/IterateExponentialParity
- D5/S1/Recurrence/Residue/IterateExponentialModSix license: citation-only triage: anchor
A396806: iterated-exponential congruences
Source and scope
Paul D. Hanna’s entry A396806 (June 9, 2026) specifies the exponential generating
series satisfying A(x) = x * exp(A^{[6]}(x)), where A^{[6]} denotes the
6-fold compositional iterate of A, not the 6-th power. The offset is 1 and
the first terms are 1, 2, 39, 1804, 139625, 15563526, 2301954109.
The conjecture proved in this repository says that a(n) is odd exactly when
n is odd, for every n >= 1. It is obtained as the instance at iterate count
6 of a statement proved uniformly in that count, alongside the defining series
equation and uniqueness among rational formal series with zero constant
coefficient.
The entry also conjectures that a(n) is congruent to n modulo three and modulo six for every n >= 1. IterateExponentialModSix.result proves the modulo-six assertion for the same aK 6 sequence, and hence the modulo-three assertion as well. Its defining equation and coefficient normalization are identified by A_equation, fixed_unique and eCoeff_encode. The inspected revision 11 still presents these assertions as conjectures; the proof is repository-derived. No A396805 residue conclusion is asserted.
Verified locator
- URL: https://oeis.org/A396806