bibkey: hanna2026a396805 authors: Paul D. Hanna year: 2026 title: “OEIS A396805: iterated-exponential congruences at iterate count 5” doi: null url: https://oeis.org/A396805 claim: “For A396805, a(n) is odd iff n is odd for n >= 1; from n = 3 its residues modulo 3 repeat 0, 1, 0.” strata_touched:
- D5/S1/Recurrence/Residue/IterateExponentialParity
- D5/S1/Recurrence/Residue/IterateExponentialFiveModThree license: citation-only triage: anchor
A396805: iterated-exponential congruences
Source and scope
Paul D. Hanna’s entry A396805 (June 9, 2026) specifies the exponential generating
series satisfying A(x) = x * exp(A^{[5]}(x)), where A^{[5]} denotes the
5-fold compositional iterate of A, not the 5-th power. The offset is 1 and
the first terms are 1, 2, 33, 1264, 80505, 7365456, 893006317.
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
5 of a statement proved uniformly in that count, alongside the defining series
equation and uniqueness among rational formal series with zero constant
coefficient.
IterateExponentialFiveModThree.result proves the source pattern [0,1,0]
modulo three starting at index three, for the same aK 5 sequence. The lower
bound is essential because a(2)=2. The earlier parity theorem remains its
separate resolution.
The source’s modulo-five assertion has recorded exact non-kernel counterexamples: a(23) is congruent to 2 while 23 is congruent to 3 modulo five, and n=24 fails as well. The present theorem does not formalize that refutation.
Verified locator
- URL: https://oeis.org/A396805