A216160 2^(2p-2) modulo p^3 for p=odd primes.

16, 6, 323, 1079, 924, 3044, 6252, 254, 21084, 4217, 42514, 48955, 63168, 101333, 90896, 87970, 164396, 100099, 85982, 221337, 464837, 90637, 214936, 735552, 171600, 330425, 437845, 311632, 363522, 1972311, 38777, 202213, 414082, 1471674, 860550, 346186

Offset

1,1

Comments

2^(4*n) == (-1)^n*(2n)!/(n!)^2 (modulo p^3) (with n = (p-1)/2) for odd primes. Except for p = 3 (n = 1), where the second expression = 25 instead of 16.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

F. Morley, Note on the Congruence 2^4n == (-1)^n*(2n)!/(n!)^2 where 2n+1 is a prime, Annals of Mathematics, Vol. 9 (1894 - 1895), pp. 168-170.

Maple

a:= proc(n) local p; p:= ithprime(n+1);
      2 &^ (2*p-2) mod p^3
    end:
seq (a(n), n=1..50);  # Alois P. Heinz, Sep 05 2012

Mathematica

Table[Mod[2^(2Prime[n] - 2), Prime[n]^3], {n, 2, 30}] (* Alonso del Arte, Sep 03 2012 *)
Table[PowerMod[2, 2p-2, p^3], {p, Prime[Range[2, 40]]}] (* Harvey P. Dale, Jun 09 2024 *)

Prog

(PARI) a(n) = { local(p); p = prime(n+1); return (2^(2*p-2) % (p^3)); }

Crossrefs

Cf. A065091.

Sequence in context: A057964 A302463 A303245 * A302371 A303087 A302892

Adjacent sequences: A216157 A216158 A216159 * A216161 A216162 A216163

Keyword

nonn

Author

Michel Marcus, Sep 03 2012

Status

approved