A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883

Offset: 1

Author

Alexander Adamchuk, Jul 01 2006

Keywords

• frac
• nonn

Comments

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...

a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Links

• Robert Israel, Table of n, a(n) for n = 1..1495
• Eric Weisstein's World of Mathematics, Prime Sums.

Crossrefs

Cf. A128646 (denominators), A119686, A006093, A000040.

Cf. A135212, A078456.

Cf. A007814, A169611, A356006.

Programs

Maple

R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025

Mathematica

Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)

PARI

a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

Formula

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007