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A122103 Sum of the fifth powers of the first n primes.

%I A122103 #20 Dec 23 2024 14:53:42
%S A122103 32,275,3400,20207,181258,552551,1972408,4448507,10884850,31395999,
%T A122103 60025150,129369107,245225308,392233751,621578758,1039774251,
%U A122103 1754698550,2599294851,3949419958,5753649309,7826720902,10903777301,14842817944
%N A122103 Sum of the fifth powers of the first n primes.
%C A122103 a(n) is prime for n = {66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, ...} = A122125. Primes of this form are listed in A122126 = {32353461605953, 9874820441996857, 10821208357045699, ...}.
%H A122103 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%H A122103 V. Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-August/011512.html">Asymptotics of sum of the first n primes with a remainder term</a>
%F A122103 a(n) = sum(k = 1 .. n, prime(k)^5).
%F A122103 a(n) =  1/6*n^6*log(n)^5 + O(n^6*log(n)^4*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). For a generalization, see comment in A122102. - _Vladimir Shevelev_, Aug 14 2013
%e A122103 a(2) = 275 because the first two primes are 2 and 3, the fifth powers of which are 32 and 243, and 32 + 243 = 275.
%e A122103 a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.
%t A122103 Table[Sum[Prime[k]^5, {k, n}], {n, 100}]
%o A122103 (PARI) a(n)=sum(i=1,n,prime(i)) \\ _Charles R Greathouse IV_, Nov 30 2013
%Y A122103 Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.
%K A122103 nonn,easy
%O A122103 1,1
%A A122103 _Alexander Adamchuk_, Aug 20 2006