%I #36 Feb 06 2015 15:45:17

%S 23,19,25,2951,25,4357,25,43,281525,269,25,37,23,295,17,3131,395191,

%T 37,25,19,139,1981,23,37,25,455,25,41,124403,61,17,511,193,535,23,

%U 5209,1951,19,25,301,891,9805,25,527,23,83,17,37,131,43,25,193,53,37,25,19

%N a(n) is the smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

%C By definition a(1) is A045345(2).

%C This sequence has a very interesting behavior. If Mod(n, 2)(Mod(n, 20)-1)(Mod(n, 20)-9)(Mod(n, 20)-13)(Mod(n, 20)-17)!=0, a(n)=17, 23 or 25; in other cases a(n) may be too large. If Mod[n, 16] = 15, a(n) = 17. For example, a(n) = 17 for n = 15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, ...; also, a(n) = 23 for n = 1, 13, 23, 35, 45, 57, 67, 89, 101, 123, 133, 145, 155, 167, 177, 189, 199, ...; a(n) = 25 for n = 3, 5, 7, 11, 19, 25, 27, 39, 43, 51, 55, 59, 65, 71, 75, ..., . For a(n) = 19 for n = 2, 20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200, 218, ..., == 2 (mod 18).

%C From _Alexander Adamchuk_, Jul 20 2008: (Start)

%C Conjectures:

%C a(n) exists for all n; a(n) >= 17.

%C a(325)-a(575) = {25,19,25,5851,1843,61,23,821,89,301,17,37,131,455,25,1607,297,37,23,19,25,

%C 325,25,37,353,47,17,1663,23,691,25,691,509269,155,25,269,105893,19,25,3971,

%C 23,213215,17,26021,327,79,25,37,151,83,23,161,101,37,25,19,327,265,17,37,25,

%C 43,23,41,169,61,25,113,21761,6289,25,47,23,19,17,4073,1137,565,25,527,25,

%C 325,25,37,23,455,25,431,13195,37,17,19,53,155,23,37,89,455,25,18839,25,6221,

%C 25,41,18597,229,17,811,623173,19,25,193,2079,673,25,881,23,47,25,37,25,97,

%C 17,79,131,37,25,19,23,56501,25,37,299,455,25,167,2707,446963,17,157,25,325,

%C 25,41,53,19,25,5917,103,1051,23,607,101,155,17,37,6233,455,25,9049,23,37,25,

%C 19,327,5359,25,37,43,455,17,9187,23,193,25,1861,7923,301,25,113,25,19,23,41,

%C 89,61,17,43,1785,131,25,37,1417,455,23,151,53,37,25,19,25,79,17,37,23,455,

%C 25,289,59,47,25,511,47,83,25,739,23,19,17,301,25,269,25,41,707,2735,23,37,

%C 299,43,25,283,69723,37,17,19,1785,479,23,37,25,455,25,1867,131,61,25,31799,

%C 23,161,17}.

%C a(n) is currently unknown and a(n)>10^7 for n = {324, 576, ...}. (End)

%C All but one of the terms up to n=1000 are known and they are less than 10^8. Currently the only unknown term for n<=1000 is a(656)>10^8. - _Alexander Adamchuk_, May 24 2009

%C a(656) > 23,491,000,000. - _Robert Price_, Apr 22 2014

%H Alexander Adamchuk and Robert Price, <a href="/A085450/b085450.txt">Table of n, a(n) for n = 1..655</a> (first 323 terms from Alexander Adamchuk)

%F For[m=2, Mod[Sum[Prime[k]^n, {k, m}], m]!=0, m++ ]; m

%e a(3) = 25 because 2^3+3^3+5^3...+prime(25)^3 == 0 (mod 25) and for 1 < n < 25 2^3+3^3+...+prime(n)^3 is not congruent to zero (mod n).

%t a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0, m++ ]; m]; Table[ a[n], {n, 1, 56}]

%t a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0&& m<10000000, m++ ]; m]; Table[ a[n], {n, 1, 1000}] (* _Alexander Adamchuk_, Jul 20 2008 *)

%o (PARI) a(n)=my(s=2^n,m=1); forprime(p=3,, if((s+=p^n)%m++==0, return(m))) \\ _Charles R Greathouse IV_, Feb 06 2015

%Y Cf. A045345, A111441, A122140, A125907, A122142, A125825, A125826, A125828, A131263, A131264, A125827, A131272, A131273, A131274, A131275, A131276, A131277, A131278, A131279.

%K nonn

%O 1,1

%A _Farideh Firoozbakht_, Aug 14 2003

%E Edited and extended by _Robert G. Wilson v_, Aug 14 2003

%E More terms: a(324) = 18642551, a(576) = 12824827. _Alexander Adamchuk_, May 24 2009