A385301 a(n) = Sum_{k=0..p-1} 1/k! mod p where p is prime(n) and 1/k! is the inverse of k! modulo p.

0, 1, 0, 3, 6, 0, 8, 4, 4, 9, 21, 0, 39, 37, 40, 32, 26, 12, 61, 6, 57, 74, 21, 41, 39, 60, 86, 64, 4, 27, 55, 2, 63, 113, 29, 42, 150, 97, 33, 84, 100, 120, 184, 72, 1, 134, 100, 78, 145, 8, 199, 98, 65, 25, 104, 95, 153, 207, 90, 132, 67, 132, 301, 251, 293, 185, 168, 176, 120, 297

Offset

1,4

Links

Robert Israel, Table of n, a(n) for n = 1..2585

Formula

a(n) = Sum_{k=0..prime(n) - 1} (k!)^(prime(n) - 2) mod prime(n) where prime(n) is the n-th prime.

a(n) = A153229(p) mod p, where p = prime(n). - David Radcliffe, Jun 26 2025

Example

a(1) = Sum_(k=0..2 - 1) (k!)^(2 - 2) mod 2 = 0.

Maple

f:= proc(n) local p, k; p:= ithprime(n); add(1/k!, k=1..p-1) mod p end proc:
map(f, [$1..100]); # Robert Israel, Jul 01 2025

Mathematica

a[n_] := Module[{p = Prime[n]}, Mod[Sum[PowerMod[k!, p-2, p], {k, 0, p-1}], p]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, prime(n) - 1, (k!)^(prime(n) - 2)) % prime(n); \\ Michel Marcus, Jun 25 2025
(Python)
from sympy import prime
def a(n):
   p = prime(n)
   s = invfact = 1
   for i in range(1, p):
       invfact = (invfact * pow(i, -1, p)) % p
       s += invfact
   return s % p # David Radcliffe, Jun 25 2025

Crossrefs

Cf. A153229, A064384.

Sequence in context: A011368 A020811 A200005 * A153097 A271854 A077086

Adjacent sequences: A385298 A385299 A385300 * A385302 A385303 A385304

Keyword

nonn

Author

Paras Dhanuka, Jun 24 2025

Extensions

More terms from Michel Marcus, Jun 25 2025

Status

approved