A055030 (Sum(m^(p-1),m=1..p-1)+1)/p as p runs through the primes.

1, 2, 71, 9596, 1355849266, 1032458258547, 1653031004194447737, 3167496749732497119310, 22841077183004879532481321652, 1768861419039838982256898243427529138091, 10293527624511391856267274608237685758691696

Offset: 1

Author

N. J. A. Sloane, Jun 11 2000

Keywords

• nonn

Comments

It is conjectured that (Sum(m^(n-1),m=1..n-1)+1)/n is an integer iff n is 1 or a prime.

Always an integer from little Fermat theorem. Converse is conjectured to be true: if p | (1+1^(p-1)+2^(p-1)+3^(p-1)+...+(p-1)^(p-1)) and p > 1, then p is prime. That was checked by Giuga up to p <= 10^1000. [Benoit Cloitre, Jun 09 2002]

For Sum(m^p, m=1..p-1)/p as p runs through the odd primes, see A219550. - Jonathan Sondow, Oct 31 2017

References

• R. K. Guy, Unsolved Problems in Number Theory, A17.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..76
• K. MacMillan and J. Sondow, Proofs of power sum and binomial coefficient congruences via Pascal's identity, Amer. Math. Monthly, 118 (2011), 549-551.

Crossrefs

Cf. A055031, A055032, A055023, A201560, A204187, A219550, A294507.

Programs

Maple

A055030 := proc(n)
    p := ithprime(n) ;
    add(m^(p-1),m=1..p-1) ;
    (1+%)/p ;
end proc:
seq(A055030(n),n=1..5) ; # R. J. Mathar, Jan 09 2017

Mathematica

Array[(Sum[m^(# - 1), {m, # - 1}] + 1)/# &@ Prime@ # &, 11] (* Michael De Vlieger, Nov 04 2017 *)

PARI

for(n=1,20,print1((1+sum(i=1, prime(n)-1,i^(prime(n)-1)))/prime(n), ",")) /* Benoit Cloitre, Jun 09 2002*/

Formula

a(n) = (1+A225578(n))/A000040(n). - R. J. Mathar, Jan 09 2017

Extensions

Comments corrected by Jonathan Sondow, Jan 11 2012