bibkey: hanna2026a396family authors: Paul D. Hanna year: 2026 title: “OEIS A396803, A396805 and A396806: iterated-exponential congruences” doi: null url: https://oeis.org/A396803 claim: “For each of A396803, A396805 and A396806, a(n) is odd iff n is odd for n >= 1; for A396803, a(n) is congruent to n modulo 3; for A396806, a(n) is congruent to n modulo 6; for A396805, residues modulo 3 repeat 0, 1, 0 from n = 3.” strata_touched:
- D5/S1/Recurrence/Residue/IterateExponentialParity
- D5/S1/Recurrence/Residue/IterateExponentialModThree
- D5/S1/Recurrence/Residue/IterateExponentialModSix
- D5/S1/Recurrence/Residue/IterateExponentialFiveModThree license: citation-only triage: anchor
Iterated-exponential family
Source and scope
Paul D. Hanna’s entries A396803 (June 8, 2026), A396805 and A396806
(June 9, 2026) specify exponential generating series satisfying
A(x) = x * exp(A^{[k]}(x)), with k = 3, 5, 6, respectively.
Here A^{[k]} denotes the k-fold compositional iterate, not the k-th power;
the zeroth iterate is the identity series x.
The shared conjecture says that a(n) is odd exactly when n is odd,
for every n >= 1. The repository proves this parity statement uniformly
in the natural iterate count, together with the defining series equation
and uniqueness among rational formal series with zero constant coefficient.
The A396803 entry also conjectures: “a(n) == [1,2,0] repeating (mod 3)
for n >= 1.” IterateExponentialModThree.result proves this statement for
the same sequence. The entry’s revision 16, retrieved September 15, 2026,
still presents it as a conjecture; the proof is repository-derived.
IterateExponentialModSix.result separately proves the A396806 congruence
a(n) = n modulo six for every positive index, hence also its modulo-three
congruence. IterateExponentialFiveModThree.result proves A396805’s pattern
0,1,0 modulo three starting at n=3; the lower bound retains the exceptional
source value a(2)=2. Its modulo-five assertion has previously recorded
non-kernel counterexamples; none of these theorems formalizes that refutation.
Verified locator
- URL: https://oeis.org/A396803
- URL: https://oeis.org/A396805
- URL: https://oeis.org/A396806