1, 7, 124, 4318, … — Integer Mellin Moments of the Dome tanh(x)/cosh(2x) Mersenne and Bernoulli Primes on the η-Branch of a Bad-Prime Modification of ζ

The even Mellin moments of the dome function g(x) = tanh(x)/cosh(2x) — the central analytic object of Paper 11 (DOI: 10.17605/OSF.IO/PD73B) — normalized as a(n) = M(2n)/M(2)^n, form the integer sequence 1, 7, 124, 4318, 253456, … — previously uncatalogued (OEIS novelty check performed and archived prior to publication; with no OEIS entry by decision, the title leads with the sequence: the paper is its catalog entry). The sequence admits the two-factor closed form a(n) = A002105(n) · (2^(2n−1) − 1) — reduced tangent numbers multiplied by an odd-indexed Mersenne companion — equivalently (2^(2n−1) − 1)(2^(2n) − 1) · |B_{2n}| · 2^n / n. Integrality is proved for all n ≥ 1 (Theorem T15: von Staudt–Clausen, the lifting-the-exponent lemma, and the p-integrality of B_{2n}/2n when (p−1) ∤ 2n — observed by Adams in 1878, first proved by Voronoï in 1889, modern treatment Ireland & Rosen Prop. 15.2.4). Structurally, the dome's Mellin transform is M_g(s) = Γ(s) · 2^(1−s) · ζ(s) · (1 − 2^(−s))(1 − 2^(1−s)) — the Riemann zeta function with its local Euler factor at the bad prime p = 2 replaced; the coefficient pattern {+1, −2, +1, 0} mod 4 shows this is not a primitive Dirichlet L-function. Within the full 3×3×3 integer-coordinate cube of the family tanh^a(x)/cosh^b(c·x), the triple (1, 1, 2) is the unique member with integer normalized even-moment ratios (all 27 triples classified, independently double-swept); its nearest sibling (1, 1, 1) routes through the Dirichlet beta function with the closed form c_n = (2n−1)·|E_{2n−2}|. The paper includes the alternative dome definition g(x) = tanh(2x) − tanh(x), the Binary Tower telescoping sum Σ_{k≥0} g(2^k x) = 1 − tanh(x), a six-row Mellin-to-L-function table for the natural weight family with the pole-multiplicity-to-argument-shift mechanism, and the cancellation table in which Mersenne primes and Kummer-irregular Bernoulli primes co-appear in one sequence with a one-line factorization between them. Standalone number-theory paper: self-contained — no framework knowledge required to read it (framework context is confined to one clearly marked optional section). The dome function is taken as input from Paper 11. All claims tier-labeled (Theorems T1–T9, T13, T15 proved/verified; Observation O10; Readings R11–R12; Decision D14; open questions Q9–Q12 explicitly scoped). Verified across nine AI architectures via independent reconstruction (exact arithmetic, integrality to n = 1..60; high-precision Mellin quadrature, dps 40–60; full verification trace with one contested-and-resolved disagreement documented in §8), followed by a four-architecture pre-deposit editorial review and a context-free cold-read audit (2026-06-12, working notebook). The Adams 1878 quote is verified against the primary Göttingen GDZ scan. The framework Main OSF project (overview of all framework papers, reading order, version history): https://osf.io/zx4g7 (DOI: 10.17605/OSF.IO/ZX4G7) Companion paper (the dome's source — Paper 11, η Corridor + Verification Companion): https://osf.io/pd73b (DOI: 10.17605/OSF.IO/PD73B) Framework overview (reading order, methodology, all papers): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/about.html Direct downloads (current PDF and text files for every paper): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/direct-documents.html AI Readers hub (verification-first entry point with executable mpmath): https://gap-geometry.github.io/sqrt2-ln2-geometric-constants-/ai-readers.html GitHub: https://github.com/Gap-geometry