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

Author

Alexander Adamchuk, Aug 20 2006

Keywords

• nonn
• easy

Comments

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, ...}.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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