Tebni's odd-prime congruence for OEIS A364822

M. Tebni's OEIS A364822, revision 25, names the sequence:

Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).

The conjecture resolved here is:

Conjectures: For p prime (p > 2), a(p) == 2 (mod p).

Preregistration: #18643. The companion conjecture is recorded in #18644.

Statement echo

The natural binder $p$ has hypotheses $p$ prime and $2<p$. The conclusion is an integer $z=a(p)$ with $z\equiv2\pmod p$. The rational formal series have coefficients $1/n!$ in even indices for cosh and in odd indices for sinh, and zero in their other indices. The coefficient $a(n)$ is $n!$ times the ordinary coefficient of their formal quotient. The existential integer equals the rational coefficient after casting; thus integrality and the congruence are both conclusions. No finite index bound or extra hypothesis is imposed.

This is an explicitly published OEIS conjecture. The settling declaration is D5.S1.Recurrence.Residue.TebniA364822.result, with admission_basis: open-problem-resolution (#18643; Proved).

The literature screen in #18643 read the OEIS entry and S. Fried's papers arXiv:2410.07237, 2606.09913 and 2607.24832; it found no A364822 proof. It also searched the Epoch OEIS Open benchmark, AlphaProof Nexus results, formal-conjectures and the repository. These are bounded negative search results; literature completeness is ASSUMED-UNVERIFIED and no priority claim is made.

Write $A(x)=\cosh(x)(1-2\sinh(x))^{-1}$. Multiplying by the denominator gives $A=\cosh+2A\sinh$. The factorial-scaled product formula IntegralEGFComposition.eCoeff_mul gives

$$ a(n)=\mathbf{1}{2\mid n}+ 2\sum{k=0}^{n-1}\binom nk a(k)\mathbf{1}_{2\nmid(n-k)}. $$

IntegralEGFComposition.integral_subst applies to the geometric series and $2\sinh(x)$, whose constant coefficient is zero. Multiplication by cosh then preserves integral factorial-scaled coefficients. The public theorem TebniA364822.num_cast states integrality at every natural index; TebniA364822.num_recurrence states the integer recurrence. At an odd prime $p$, the cosh term is zero and all $0<k<p$ binomial terms vanish modulo $p$. The term $k=0$ is $1$, so the factor $2$ gives the required residue.

An odd prime $p>2$ with nonintegral $a(p)$ or $a(p)\not\equiv2\pmod p$ would refute the claim.

The Lean source is the settling module, with its Blueprint mirror. The source note is tebni2023a364822. The public result proves the full proposition, including its integer witness. The module uses only the standard axiom allowance.

Four-slot escape auditing is unfinished for TebniA364822.num_cast, TebniA364822.num_recurrence, TebniA364822.result and TebniA364822PowerOfTwo.result; the concrete template gap and missing realization/source-selection evidence are recorded in issue #22441. No registration is asserted to be declared_validated.

Experiment

The independent computation is the pinned experiment. Its script check.py has SHA-256 52f51c8b6cb41d6525e4b991d6c081a96f6bd9ff9783c9ab6888857e2919ce24. The command python3 check.py 256 100001, using the script from that revision, exited $0$ with all reported checks true. Its tested scopes are odd primes at most $256$, powers $2^m\le256$ with $0\le m\le8$, every even $2\le n\le256$ for the exact valuation, and odd $5\le r\le100001$ for the parity count. The Taylor expansion is checked for $0\le n<16$, the second expansion for $0\le n<40$, and the published coefficient list for $0\le n<12$. The script's coefficient function, evaluated at $n=2$, gives $a(2)=9$ and $a(2)\bmod2=1$. This evaluation exits $0$ using the same script hash.

• Proved, kernel-checked: every $a(n)$ is an integer, by TebniA364822.num_cast; this includes $n=1$ and the modulus-one case of TebniA364822PowerOfTwo.result at $m=0$.
• Proved, kernel-checked: at an odd prime, the interior binomial terms vanish and the constant-index term gives residue $2$; TebniA364822.result uses Nat.Prime.dvd_choose_self for this step.
• Proved, kernel-checked: if $k$ is odd then $2^m\mid\binom{2^m}{k}$, by the private helper TebniA364822PowerOfTwo.pow_two_dvd_choose_odd. TebniA364822PowerOfTwo.result applies it to the recurrence for positive $m$ and handles $m=0$ modulo one.
• Computed: the experiment above validates both congruences through $n\le256$, $v_2(a(n)-1)=v_2(n)+2$ for every even $2\le n\le256$, and its parity count for odd $5\le r\le100001$. The cited command, exit code, immutable revision and SHA-256 specify the evidence.
• Computed: $a(2)=9$ has residue $1$ modulo $2$, by the same script's coefficient function. Thus the prime $2$ cannot be included in the residue-$2$ assertion.
• Open: the sharper exact valuation for every even $n\ge2$ is unformalized here. The finite experiment does not prove its uniform extension.

What the settlement shows

The decisive mechanism is the exact integer binomial recurrence. It proves both published congruences without the sharper valuation assertion. Integrality extends to all natural indices, not just primes or powers of two. The strict prime bound is necessary, as the computed $p=2$ value shows. For powers of two the zero exponent is included and gives modulus one. The residue-$1$ conclusion modulo $2^m$ is kernel-checked; the stronger modulus $2^{m+2}$ and the exact valuation at arbitrary even indices remain open as formalization targets. The source lists these two conjectures; their Lean proofs share the recurrence, and the dyadic proof does not use the odd-prime settling result as a premise.

Completeness of the literature screen is ASSUMED-UNVERIFIED. The uniform exact valuation is not a kernel-checked result of these modules. Escape registration remains unfinished under issue #22441.