cp's OEIS Frontend

A097878 Decimal expansion of Sum_{k>=1} k/prime(k)^4.

%I A097878 #20 Nov 07 2022 20:27:21
%S A097878 0,9,4,4,4,1,8,5,8,1,9,6,5,0,4,9,4,2,1,8,4
%N A097878 Decimal expansion of Sum_{k>=1} k/prime(k)^4.
%C A097878 From _Jon E. Schoenfield_, Nov 07 2022: (Start)
%C A097878 Let M = 10^10, and let J be the number of primes < M, i.e., J = pi(M) = 455052511; then prime(J+1) = 10000000019.
%C A097878 Since prime(J+1) > M+2 and prime(k+1) - prime(k) >= 2 for all k > 1, it follows that, for all k > J,
%C A097878    prime(k) > M + 2*(k - J)
%C A097878 and thus
%C A097878    k/prime(k)^4 < k/(M + 2*(k - J))^4
%C A097878 so
%C A097878    Sum_{k>J} k/prime(k)^4 < Sum_{k>J} k/(M + 2*(k - J))^4
%C A097878 and it can be shown that the sum on the right-hand side is a value < 5*10^-22.
%C A097878 Summing the values of k/prime(k)^4 for all k <= J to obtain
%C A097878    Sum_{k=1..J} k/prime(k)^4 = 0.0944418581965049421841...
%C A097878 yields a lower bound on the infinite sum, and since the infinite sum is
%C A097878    Sum_{k>=1} k/prime(k)^4 = Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(k)^4,
%C A097878 it must be less than
%C A097878    Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/(M + 2*(k - J))^4,
%C A097878 which is less than
%C A097878    0.0944418581965049421842 + 5*10^-22 = 0.0944418581965049421847,
%C A097878 which thus provides an upper bound on the infinite sum. (End)
%e A097878 0.094441858196504942184...
%Y A097878 Cf. A097906, A097879.
%K A097878 more,nonn,cons
%O A097878 0,2
%A A097878 _Pierre CAMI_, Sep 02 2004
%E A097878 a(15)-a(17) corrected and a(18)-a(21) added by _Jon E. Schoenfield_, Nov 07 2022