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

0, 9, 4, 4, 4, 1, 8, 5, 8, 1, 9, 6, 5, 0, 4, 9, 4, 2, 1, 8, 4

Offset: 0

Author

Pierre CAMI, Sep 02 2004

Keywords

• more
• nonn
• cons

Comments

From Jon E. Schoenfield, Nov 07 2022: (Start)

Let M = 10^10, and let J be the number of primes < M, i.e., J = pi(M) = 455052511; then prime(J+1) = 10000000019.

Since prime(J+1) > M+2 and prime(k+1) - prime(k) >= 2 for all k > 1, it follows that, for all k > J,

prime(k) > M + 2*(k - J)

and thus

k/prime(k)^4 < k/(M + 2*(k - J))^4

so

Sum_{k>J} k/prime(k)^4 < Sum_{k>J} k/(M + 2*(k - J))^4

and it can be shown that the sum on the right-hand side is a value < 5*10^-22.

Summing the values of k/prime(k)^4 for all k <= J to obtain

Sum_{k=1..J} k/prime(k)^4 = 0.0944418581965049421841...

yields a lower bound on the infinite sum, and since the infinite sum is

Sum_{k>=1} k/prime(k)^4 = Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/prime(k)^4,

it must be less than

Sum_{k=1..J} k/prime(k)^4 + Sum_{k>J} k/(M + 2*(k - J))^4,

which is less than

0.0944418581965049421842 + 5*10^-22 = 0.0944418581965049421847,

which thus provides an upper bound on the infinite sum. (End)

Examples

			0.094441858196504942184...
		

Crossrefs

Cf. A097906, A097879.

Extensions

a(15)-a(17) corrected and a(18)-a(21) added by Jon E. Schoenfield, Nov 07 2022