A348883 Number of base-p digits for which the Apéry numbers support a Lucas congruence modulo p^2, where p is the n-th prime.

2, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 9, 3, 3, 5, 5, 3, 5, 3, 5, 3, 3, 7, 3, 7, 3, 3, 3, 3, 5, 3, 3, 3, 7, 3, 3, 5, 3, 5, 5, 3, 7, 3, 3, 3, 5, 3, 3, 3, 5, 5, 3, 5, 3, 7, 5, 3, 7, 7, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 3, 3

Offset

1,1

Comments

Gessel showed that A005259(p m + d) = A005259(m) * A005259(d) mod p for all m >= 0 and all d in the range 0 <= d <= p - 1. a(n) is the number of such d for which this congruence also holds modulo p^2 for all m >= 0, where p is the n-th prime.

Equivalently, a(n) is the number of integers d in the range 0 <= d <= p - 1 such that A005259(d) = A005259(p - 1 - d) mod p^2, where p is the n-th prime.

Links

Eric Rowland, Table of n, a(n) for n = 1..256

Ira Gessel, Some congruences for Apéry numbers, Journal of Number Theory 14 (1982) 362-368.

Eric Rowland, Reem Yassawi, and Christian Krattenthaler, Lucas congruences for the Apéry numbers modulo p^2, Integers 21 (2021) A20.

Example

The 4th prime is 7, and there are a(4) = 5 base-7 digits d such that A005259(7 m + d) = A005259(m) * A005259(d) mod 7^2 for all m >= 0, namely 0, 2, 3, 4, and 6. Equivalently, these are the 5 base-7 digits d satisfying A005259(d) = A005259(6 - d) mod 7^2:
    A005259(0) = 1 = 21460825 = A005259(6) mod 7^2
    A005259(2) = 73 = 24 = 33001 = A005259(4) mod 7^2
    A005259(3) = 1445 = 24 = 1445 = A005259(3) mod 7^2

Mathematica

A005259[n_] := Sum[Binomial[n, k]^2 Binomial[n + k, k]^2, {k, 0, n}]
Table[If[p == 2, 2, 1 + 2 Length[Select[Range[0, (p - 3)/2], Mod[A005259[#], p^2] == Mod[A005259[p - 1 - #], p^2] &]]], {p, Prime[Range[20]]}]

Crossrefs

Cf. A005259.

A348884 is the analogous sequence for binomial coefficients.

Sequence in context: A252502 A063256 A229703 * A131320 A020483 A119912

Adjacent sequences: A348880 A348881 A348882 * A348884 A348885 A348886

Keyword

nonn

Author

Eric Rowland, Nov 02 2021

Status

approved