A122102 a(n) = Sum_{k=1..n} prime(k)^4.

16, 97, 722, 3123, 17764, 46325, 129846, 260167, 540008, 1247289, 2170810, 4044971, 6870732, 10289533, 15169214, 23059695, 35177056, 49022897, 69174018, 94585699, 122983940, 161934021, 209392342, 272134583, 360663864, 464724265, 577275146, 708354747, 849512908

Offset: 1

Author

Alexander Adamchuk, Aug 20 2006

Keywords

• nonn

Comments

a(n) is prime for n = {2,32,90,110,134,152,168,180,194,...} = A122127.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• 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. A007504, A024450, A098999, A122103, A122127.

Partial sums of A030514.

Programs

Magma

[&+[NthPrime(k)^4: k in [1..n]]: n in [1..30]]; // G. C. Greubel, Oct 02 2019

Maple

seq(add(ithprime(k)^4, k=1..n), n=1..30); # G. C. Greubel, Oct 02 2019

Mathematica

Table[Sum[Prime[k]^4,{k,1,n}],{n,1,100}]
Accumulate[Prime[Range[30]]^4] (* Harvey P. Dale, Aug 07 2021 *)

PARI

a(n)=my(s);forprime(p=2,prime(n),s+=p^4); s \\ Charles R Greathouse IV, Aug 02 2013

Sage

[sum(nth_prime(k)^4 for k in (1..n)) for n in (1..30)] # G. C. Greubel, Oct 02 2019

Formula

From Vladimir Shevelev, Aug 02 2013: (Start)

a(n) = 0.2*n^5*log(n)^4 + O(n^5*log(n)^3*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev).

A generalization: Sum_{i=1..n} prime(i)^k = 1/(k+1)*n^(k+1)*log(n)^k + O(n^(k+1)*log(n)^(k-1)*log(log(n))).

(End)