A122126 -id:A122126 - cp's OEIS frontend

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

32, 275, 3400, 20207, 181258, 552551, 1972408, 4448507, 10884850, 31395999, 60025150, 129369107, 245225308, 392233751, 621578758, 1039774251, 1754698550, 2599294851, 3949419958, 5753649309, 7826720902, 10903777301, 14842817944

Offset: 1

Alexander Adamchuk, Aug 20 2006

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, ...}.

			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.
a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.

OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.

Table[Sum[Prime[k]^5, {k, n}], {n, 100}]

a(n)=sum(i=1,n,prime(i)) \\ Charles R Greathouse IV, Nov 30 2013

a(n) = sum(k = 1 .. n, prime(k)^5).

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

A122125 Indices n such that A122103[n] = Sum[ Prime[k]^5, {k,1,n}] is a prime.

Original entry on oeis.org

66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, 1002, 1030, 1034, 1038, 1064, 1086, 1092, 1108, 1162, 1322, 1394, 1424, 1468, 1500, 1630, 1654, 1668, 1684, 1694, 1840, 1886, 1980

Offset: 1

Alexander Adamchuk, Aug 20 2006

nonn

Corresponding primes of form Sum[ Prime[k]^5, {k,1,n} ] are listed in A122126[n] = {32353461605953,9874820441996857,10821208357045699,...}.

Cf. A122103, A122126.

Select[Range[2000],PrimeQ[Sum[Prime[k]^5,{k,1,#1}]]&]

cp's OEIS Frontend

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

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