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A232822 Prime(m), where m is such that (Sum_{k=1..m} prime(k)^8) / m is an integer.

%I A232822 #45 Dec 03 2024 08:14:41
%S A232822 2,191,12599173,53029063,22806625723729,27568116247823,41455846079203,
%T A232822 289700908580893,1194728983756489,6275148480751847
%N A232822 Prime(m), where m is such that (Sum_{k=1..m} prime(k)^8) / m is an integer.
%C A232822 The primes correspond to indices m = 1, 43, 824747, 3171671, ... = A125828. - _M. F. Hasler_, Dec 01 2013
%C A232822 a(10) > 1352363608564489. - _Bruce Garner_, Jul 07 2021
%C A232822 a(11) > 18205684894350047. - _Paul W. Dyson_, Dec 03 2024
%H A232822 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%F A232822 a(n) = prime(A125828(n)). - _M. F. Hasler_, Dec 01 2013
%e A232822 a(2) = 191, because 191 is the 43rd prime and the sum of the first 43 primes^8 = 7287989395992721002 = 43 * 169488125488202814.
%t A232822 t = {}; sm = 0; Do[sm = sm + Prime[n]^8; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
%o A232822 (PARI) is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^8); s==0 \\ _Charles R Greathouse IV_, Nov 30 2013
%o A232822 (PARI) S=n=0;forprime(p=1,,(S+=p^8)%n++||print1(p",")) \\ _M. F. Hasler_, Dec 01 2013
%Y A232822 Cf. A125828.
%Y A232822 Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
%Y A232822 Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
%Y A232822 Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
%K A232822 nonn,more
%O A232822 1,1
%A A232822 _Robert Price_, Nov 30 2013
%E A232822 a(5)-a(6) from _Paul W. Dyson_, Jan 01 2021
%E A232822 a(7) from _Bruce Garner_, Mar 02 2021
%E A232822 a(8) from _Bruce Garner_, Mar 30 2021
%E A232822 a(9) from _Bruce Garner_, Jul 07 2021
%E A232822 a(10) from _Paul W. Dyson_, Jul 07 2023