A347310 a(n) = smallest k such that Sum_{i=1..k} log(p_i)/p_i >= n, where p_i is the i-th prime.

3, 8, 19, 43, 100, 236, 562, 1354, 3300, 8119, 20136, 50302, 126451, 319628, 811829, 2070790, 5302162, 13621745, 35101258, 90696900, 234924747, 609864582, 1586430423, 4134442382, 10793331294, 28221407514, 73898377351

Offset

1,1

Comments

Suggested by Mertens's theorem that Sum_{p <= x} log(p)/p = log(x) + O(1).

References

Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Society, 2015. See page 16.

Links

Table of n, a(n) for n=1..27.

Formula

a(n) = pi(A347311(n)) = A000720(A347311(n)). - Michel Marcus, Sep 06 2021

Example

a(1) = 3 because log(2)/2 + log(3)/3 + log(5)/5 = 1.034665268989... is the first time the sum is >= 1.

Mathematica

Table[k=1; While[Sum[Log@Prime@i/Prime@i, {i, ++k}]<n]; k, {n, 8}] (* Giorgos Kalogeropoulos, Sep 08 2021 *)

Prog

(PARI) a(n) = my(k=0, s=0, p=2); while (s < n, s += log(p)/p; k++; p = nextprime(p+1)); k; \\ Michel Marcus, Sep 06 2021

Crossrefs

Cf. A000720, A002387, A016088, A046024, A347311.

Sequence in context: A054480 A371796 A191787 * A332719 A326599 A121551

Adjacent sequences: A347307 A347308 A347309 * A347311 A347312 A347313

Keyword

nonn,more

Author

N. J. A. Sloane, Sep 06 2021

Extensions

a(8)-a(16) from Michel Marcus, Sep 06 2021

a(17)-a(23) from Jon E. Schoenfield, Sep 06 2021

a(24)-a(27) from Amiram Eldar, Sep 10 2024

Status

approved