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A232733 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^6) / m is an integer.

%I A232733 #62 Jan 17 2025 22:32:24
%S A232733 2,41647,3197891,630397289,779089704751,3819383648849,44041722668737,
%T A232733 1322879640047263,9863536132182127,16069251644649407,32520030920151967
%N A232733 Prime(m), where m is such that (Sum_{i=1..m} prime(i)^6) / m is an integer.
%C A232733 The primes correspond to indices n = 1, 4357, 230065, 32826947, 29578097627 = A125825.
%C A232733 a(12) > 3.7*10^16. - _Paul W. Dyson_, Jan 17 2025
%H A232733 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F A232733 a(n) = prime(A125828(n)).
%e A232733 a(2) = 41647, because 41647 is the 4357th prime and the sum of the first 4357 primes^6 = 2952411812082729747782733271068 when divided by 4357 equals 677624928180566845945084524 which is an integer.
%t A232733 t = {}; sm = 0; Do[sm = sm + Prime[n]^6; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%o A232733 (PARI) is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^6); s==0 \\ _Charles R Greathouse IV_, Nov 30 2013
%o A232733 (PARI) S=n=0;forprime(p=1,,(S+=p^6)%n++||print1(p",")) \\ _M. F. Hasler_, Dec 01 2013
%Y A232733 Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
%Y A232733 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A232733 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A232733 nonn,more
%O A232733 1,1
%A A232733 _Robert Price_, Dec 02 2013
%E A232733 a(6) from _Bruce Garner_, Jul 10 2021
%E A232733 a(7) from _Paul W. Dyson_, Jan 08 2021
%E A232733 a(8) from _Bruce Garner_, Jul 10 2021
%E A232733 a(9) from _Paul W. Dyson_, Oct 21 2022
%E A232733 a(10) from _Paul W. Dyson_, Oct 31 2022
%E A232733 a(11) from _Paul W. Dyson_, Dec 08 2022