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%I #37 Feb 04 2022 11:18:20
%S 1,25,837,5129,94375,271465,3576217,3661659,484486719,2012535795,
%T 31455148645,95748332903,145967218799,165153427677,21465291596581,
%U 97698929023845
%N Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).
%C No other terms up to 10^8. - _Stefan Steinerberger_, Jun 06 2007
%C a(11) > 6*10^9. - _Donovan Johnson_, Oct 15 2012
%C a(13) > 10^11. - _Robert Price_, Mar 30 2013
%C a(15) > 10^12. - _Paul W. Dyson_, Jan 04 2021
%C a(16) > 2.2*10^13. - _Bruce Garner_, May 09 2021
%C a(17) > 10^14. - _Paul W. Dyson_, Feb 04 2022
%H OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%e a(2) = 25 because 25 is the first number n>1 that divides A122103[n] = Sum[ Prime[k]^5, {k,1,n} ].
%e Mod[ A122103[25], 25] = Mod[ 2^5 + 3^5 + 5^5 + ... + 89^5 + 97^5, 25 ] = 0.
%t s = 0; t = {}; Do[s = s + Prime[n]^5; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t
%t Module[{nn = 4*10^6},Select[Thread[{Range[nn], Accumulate[ Prime[ Range[ nn]]^5]}], Divisible[#[[2]], #[[1]]] &]][[All, 1]] (* Generates the first 8 terms; to generate more, increase the value of nn, but the program may take a long time to run. *) (* _Harvey P. Dale_, Aug 26 2019 *)
%Y Cf. A122103, A098999, A007504, A045345, A024450, A111441, A122102, A122140.
%K nonn,hard
%O 1,2
%A _Alexander Adamchuk_, Aug 21 2006
%E 2 more terms from _Stefan Steinerberger_, Jun 06 2007
%E a(9)-a(10) from _Donovan Johnson_, Oct 15 2012
%E a(11)-a(12) from _Robert Price_, Mar 30 2013
%E a(13)-a(14) from _Paul W. Dyson_, Jan 04 2021
%E a(15) from _Bruce Garner_, May 09 2021
%E a(16) from _Paul W. Dyson_, Feb 04 2022
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