A097906 Decimal expansion of Sum_{i >= 1} i/prime(i)^2.

1, 1, 4, 9, 0, 6, 4, 1, 7

Offset: 1

Author

Pierre CAMI, Sep 04 2004, Dec 04 2014

Keywords

• cons
• more
• nonn

Comments

By the prime number theorem the growth of n/prime(n)^2 is of the same order as 1/(n*log(n)^2). So the convergence of this series is as slow as the convergence of A115563. The current vague estimate is derived solely from the brute force accumulation of the partial sums (see Examples).

For the first 10000000 primes the sum = 1.0949524507.

For the first 20000000 primes the sum = 1.0970205319.

For the first 30000000 primes the sum = 1.0981565164.

For the first 40000000 primes the sum = 1.0989320390.

For the first 50000000 primes the sum = 1.0995170340.

For the first 60000000 primes the sum = 1.0999846910.

For the first 70000000 primes the sum = 1.1003730572.

For the first 80000000 primes the sum = 1.1007044057.

For the first 90000000 primes the sum = 1.1009928516.

For the first 100000000 primes the sum = 1.1012478922.

However, this does not take into account the non-negligible contribution from the tail.

From Charles R Greathouse IV, Dec 04 2014: (Start)

My current estimate of the constant is 1.1490642... based on a summation of the primes up to 10^10, 1.10463086777...

plus the integral from pi(10^10) + 1/2 to infinity of x/ali(x)^2 where ali(x) is the inverse logarithmic integral, which contributes 0.04443338335... (End)

Examples

			1.14906417...
		

Crossrefs

Cf. A115563, A085548.

Extensions

Corrected and extended by Charles R Greathouse IV, Dec 04 2014

Extended by David Broadhurst, Dec 07 2014