A253357 Decimal expansion of Sum_{n>=1} prime(n)/n^4.

2, 3, 3, 7, 6, 3, 5, 3, 2, 9, 7, 4

Offset

1,1

Comments

Since prime(n) ~ n*log(n), Sum_{n >=1} prime(n)/n^j converges only when j > 2.

The partial sum over n <= 100000 is 2.33763532906803560, over n <= 1000000 is 2.337635329736982..., over n <= 5000000 is 2.3376353297446376... and over n <= 10000000 is 2.33763532974490000... - R. J. Mathar, Nov 05 2015

Links

Table of n, a(n) for n=1..12.

Olivier Izad, Java program

Example

2.3376353297449...
The first few iterations of the sum are:
n=1, 2/1^4 = 2;
n=2, 2 + 3/2^4 = 2.1875;
n=3, 2 + 3/2^4 + 5/3^4 = 2.249...

Mathematica

s = 0; k = 1; p = 2; While[k < 100000001, s = N[s + p/k^4, 24]; k++; p = NextPrime@ p]; s (* Robert G. Wilson v, Jan 27 2015 *)

Prog

(bash)
awk 'BEGIN{n=1}; {sum=sum+$1/(n^4); n++; OFMT="%.50f"; print sum}' primes.txt
(Java) See attachment
(PARI) suminf(n=1, prime(n)/n^4) \\ Michel Marcus, Jan 10 2015

Crossrefs

Cf. A253358 (sum over prime(n)/n^3).

Sequence in context: A228548 A140407 A063670 * A185909 A347533 A193713

Adjacent sequences: A253354 A253355 A253356 * A253358 A253359 A253360

Keyword

nonn,cons,more

Author

Olivier Izad, Dec 30 2014

Status

approved