A161527 Numerators of cumulative sums of rational sequence A038110(k)/A038111(k).

1, 2, 11, 27, 61, 809, 13945, 268027, 565447, 2358365, 73551683, 2734683311, 112599773191, 4860900544813, 9968041656757, 40762420985117, 83151858555707, 5085105491885327, 341472595155548909, 24295409051193284539, 1777124696397561611347

Offset: 1

Author

Daniel Tisdale, Jun 12 2009

Keywords

• nonn
• frac

Comments

By rewriting the sequence of sums as 1 - Product_{n>=1} (1 - 1/prime(n)), one can show that the product goes to zero and the sequence of sums converges to 1. This is interesting because the terms approach 1/(2*prime(n)) for large n, and a sum of such terms might be expected to diverge, since Sum_{n>=1} 1/(2*prime(n)) diverges.

Denominators appear to be given by A060753(n+1). - Peter Kagey, Jun 08 2019

A254196 appears to be a duplicate of this sequence. - Michel Marcus, Aug 05 2019

Links

• Peter Kagey, Table of n, a(n) for n = 1..400

Crossrefs

Cf. A038110, A038111, A060753, A254196.

Programs

Mathematica

Numerator[Table[1 - Product[1 - (1/Prime[k]), {k,1,n}], {n,1,20}]]

PARI

r(n) = prod(k=1, n-1, (1 - 1/prime(k)))/prime(n);
a(n) = numerator(sum(k=1, n, r(k))); \\ Michel Marcus, Jun 08 2019

Formula

a(n) = A053144(n)/A058250(n). - Jamie Morken, Aug 28 2022