A383024 a(n) = floor(1/(1-s(n))) where s(n) = Sum_{k=1..n} phi(k)/k * log(2^k/(2^k-1)). phi(k) is the Euler totient function A000010(k).

3, 6, 13, 23, 61, 90, 229, 417, 917, 1429, 3916, 5748, 16323, 28489, 53121, 89322, 249079, 364536, 1068096, 1802530, 3542367, 5749605, 16694858, 24980973, 61771834, 107398135, 230197552, 363961698, 1053601058, 1426995284, 3997869809, 7478431118

Offset

1,1

Comments

The infinite series {s(n)} converges to 1 from below (see link).

Links

Table of n, a(n) for n=1..32.

Yifan Xie, The asymptotics of a convergent series

Formula

2^n <= a(n) < 2^(n^2).

Example

For n = 3, s(3) = log(16/(sqrt(3)*7^(2/3))) = 0.926009..., so a(3) = floor(1/(1-0.926009...)) = 13.

Prog

(PARI) v = [log(2)]; for(n=2, 32, v = concat(v, v[n-1] + eulerphi(n)/n*log(2^n/(2^n-1)))); a(n) = floor(1/(1-v[n])); \\ can only compute 63 terms due to loss of precision

Crossrefs

Cf. A000010.

Sequence in context: A323580 A002799 A285263 * A162426 A374627 A058554

Adjacent sequences: A383021 A383022 A383023 * A383025 A383026 A383027

Keyword

nonn

Author

Yifan Xie, Apr 13 2025

Status

approved