A046024 a(n) = smallest k such that Sum_{ i = 1..k } 1/prime(i) exceeds n.

1, 3, 59, 361139, 43922730588128390

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• more
• nice

Comments

The corresponding primes prime(a(n)) are in A016088.

Index m for which the prime harmonic number p[ m ] := Sum[ 1/Prime[ k ],{k,1,m} ] >= n.

Links

• E. Bach, D. Klyve, and J. P. Sorenson, Computing prime harmonic sums, Math. Comp. 78 (2009) 2283-2305.
• Eric Weisstein's World of Mathematics, Prime Number.
• Eric Weisstein's World of Mathematics, Harmonic Series of Primes.

Crossrefs

Cf. A004080, A016088, A096232, A223037.

Cf. A024451/A002110 for Sum_{i = 1..n} 1/prime(i).

Programs

Mathematica

Table[m = 1; s = 0; While[(s = s + 1/Prime[m]) <= n, m++];
m, {n, 0, 4}] (* Robert Price, Mar 27 2019 *)

PARI

a(n)=my(t); forprime(p=2,, t+=1./p; if(t>n, return(primepi(p)))) \\ Charles R Greathouse IV, Apr 29 2015

Formula

From Jonathan Sondow, Apr 17 2013: (Start)

a(n) = A000720(A016088(n)) = A000720(A096232(n))+1.

a(n) = e^(e^(n + O(1))), see comment in A223037. [corrected by Charles R Greathouse IV, Aug 22 2013] (End)

a(n) = A103591(2*n). - Michel Marcus, Aug 22 2013

Extensions

a(4) found by Tomás Oliveira e Silva (tos(AT)det.ua.pt), using the fourth term of A016088. - Dec 14 2005

a(0) from Jonathan Sondow, Apr 16 2013