A383024 - OEIS

%I #10 Apr 18 2025 21:25:29

%S 3,6,13,23,61,90,229,417,917,1429,3916,5748,16323,28489,53121,89322,

%T 249079,364536,1068096,1802530,3542367,5749605,16694858,24980973,

%U 61771834,107398135,230197552,363961698,1053601058,1426995284,3997869809,7478431118

%N 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).

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

%H Yifan Xie, <a href="/A383024/a383024_1.pdf">The asymptotics of a convergent series</a>

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

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

%o (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

%Y Cf. A000010.

%K nonn

%O 1,1

%A _Yifan Xie_, Apr 13 2025