A254196 a(n) is the numerator of Product_{i=1..n} (1/(1-1/prime(i))) - 1.

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

Offset: 1

Author

Geoffrey Critzer, Jan 26 2015

Keywords

• nonn

Comments

The denominators are A038110(n+1).

a(n)/A038110(n+1) = Sum_{k >=2} 1/k where k is a positive integer whose prime factors are among the first n primes. In particular, for n=1,2,3,4,5, a(n)/A038110(n+1) is the sum of the reciprocals of the terms (excepting the first, 1) in A000079, A003586, A051037, A002473, A051038.

Appears to be a duplicate of A161527. - Michel Marcus, Aug 05 2019

Examples

			a(1)=1 because 1/2 + 1/4 + 1/8 + 1/16 + ... = 1/1.
a(2)=2 because 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + ... = 2/1.
a(3)=11 because 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + ... = 11/4.
a(4)=27 because Sum_{n>=2} 1/A002473(n) = 27/8.
a(5)=61 because Sum_{n>=2} 1/A051038(n) = 61/16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..422
• Eric Weisstein's World of Mathematics, Smooth Number

Crossrefs

Cf. A038110, A000079, A003586, A051037, A002473, A051038, A161527.

Programs

Maple

seq(numer(mul(1/(1-1/ithprime(i)),i=1..n)-1),n=1..20); # Robert Israel, Jan 28 2015

Mathematica

Numerator[Table[Product[1/(1 - 1/p), {p, Prime[Range[n]]}] - 1, {n,1,20}]]
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Numerator@ Table[b[n], {n, 1, 20}] (* Fred Daniel Kline, Jun 27 2017 *)

PARI

a(n) = numerator(prod(i=1, n, (1/(1-1/prime(i)))) - 1); \\ Michel Marcus, Jun 29 2017

Formula

a(n) = A038111(n+1)/prime(n+1)-A038110(n+1). - Robert Israel, Jan 28 2015, corrected Jul 07 2019.