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A055030 (Sum(m^(p-1),m=1..p-1)+1)/p as p runs through the primes. 10
1, 2, 71, 9596, 1355849266, 1032458258547, 1653031004194447737, 3167496749732497119310, 22841077183004879532481321652, 1768861419039838982256898243427529138091, 10293527624511391856267274608237685758691696 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

FORMULA

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

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A221959 A221553 A071871 * A185120 A217842 A053318

Adjacent sequences:  A055027 A055028 A055029 * A055031 A055032 A055033

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 11 2000

EXTENSIONS

Comments corrected by Jonathan Sondow, Jan 11 2012

STATUS

approved

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Last modified April 6 20:06 EDT 2020. Contains 333286 sequences. (Running on oeis4.)