A385820 Number of equivalence classes of finitely-supported integer functions on Z^2 modulo moves that add + or -1 to every cell whose coordinates form an arithmetic progression of length n.

1, 2, 27, 1024, 9765625, 272097792, 558545864083284007, 295147905179352825856, 1144561273430837494885949696427, 305175781250000000000000000000000000, 1890591424712781041871514584574319778449301246603238034051, 98746073676238604311280222171685832518740805156864

Offset

1,2

Links

Ethan Ji, Table of n, a(n) for n = 1..36

Formula

a(n) = Product_{p^k | n : prime p, k = p-adic order of n} p^(n^2*(k*p^(2k) - p^k(p^k - 1)/(p - 1)) / (2*p^(2k))).

a(p) = A076113(p), for prime p.

a(n) = Product_{k=1..n} (n/gcd(n,k))^k. - Natalia L. Skirrow, Jan 11 2026

Mathematica

a[n_Integer?Positive] := Module[{pairs = FactorInteger[n]}, Times @@ (#1^(n^2*(#2 #1^(2 #2) - (#1^#2 (#1^#2 - 1))/(#1 - 1))/(2 #1^(2 #2))) & @@@ pairs)]

Prog

(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1], k=f[i, 2]); f[i, 2] = n^2*(k*p^(2*k) - p^k*(p^k-1)/(p-1))/(2*p^(2*k))); factorback(f); \\ Michel Marcus, Jul 10 2025
(Python) from math import gcd, prod
A385820=lambda n: prod(map(lambda k: (n//gcd(n, k))**k, range(1, n))) # Natalia L. Skirrow, Jan 11 2026

Crossrefs

Cf. A076113.

Sequence in context: A078102 A395106 A221534 * A395074 A221535 A067075

Adjacent sequences: A385817 A385818 A385819 * A385821 A385822 A385823

Keyword

nonn

Author

Ethan Ji, Jul 09 2025

Status

approved