A226963 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 5 (mod n).

1, 2, 5, 10, 30, 210, 9030, 235290, 11072512110

Offset: 1

Author

José María Grau Ribas, Jun 24 2013

Keywords

• nonn
• more

Comments

Also, numbers n such that B(n)*n == 5 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -5 (mod n). - Max Alekseyev, Aug 26 2013

There are no other terms below 10^31. - Max Alekseyev, Apr 04 2018

Links

• M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT]

Crossrefs

Cf. A031971.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), A226962 (m=4), this sequence (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Programs

Mathematica

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == 5 &]

PARI

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-5 \\ Charles R Greathouse IV, Nov 13 2013

Extensions

Terms 1,2,5 prepended and a(9) added by Max Alekseyev, Aug 26 2013