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Numerators of cumulative sums of rational sequence A038110(k)/A038111(k).
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%I #61 Oct 09 2022 21:22:53

%S 1,2,11,27,61,809,13945,268027,565447,2358365,73551683,2734683311,

%T 112599773191,4860900544813,9968041656757,40762420985117,

%U 83151858555707,5085105491885327,341472595155548909,24295409051193284539,1777124696397561611347

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

%C 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.

%C Denominators appear to be given by A060753(n+1). - _Peter Kagey_, Jun 08 2019

%C A254196 appears to be a duplicate of this sequence. - _Michel Marcus_, Aug 05 2019

%H Peter Kagey, <a href="/A161527/b161527.txt">Table of n, a(n) for n = 1..400</a>

%F a(n) = A053144(n)/A058250(n). - _Jamie Morken_, Aug 28 2022

%t Table [1- Product[1 - (1/Prime[k])), {i,1,j}, {j,1,20}]; (* This is a table of the individual sums: Sum[Product[ 1 - (1/Prime[k]),{k,n-1}]/Prime[n],{n,1,3}], which is the sum of terms of the Mathematica table given in A038111 (three terms, in this example). *)

%o (PARI) r(n) = prod(k=1, n-1, (1 - 1/prime(k)))/prime(n);

%o a(n) = numerator(sum(k=1, n, r(k))); \\ _Michel Marcus_, Jun 08 2019

%Y Cf. A038110, A038111, A060753, A254196.

%K nonn,frac

%O 1,2

%A _Daniel Tisdale_, Jun 12 2009