A089363 - OEIS

%I #24 Jan 14 2025 03:15:55

%S 3,16,50,145,407,1177,3508,10677,32967,102719,321798,1011538,3186390,

%T 10050746,31730137,100228044,316713624,1001037551,3164497350,

%U 10004755379,31632975601,100021893197,316274794667,1000101078155,3162495003354,10000467510250,31623782520067

%N Numbers of pairs (i, j), i, j > 1, such that i^j <= 10^n.

%C These numbers are related to the divergent series Sum_{k=2..r} n^(1/k) = n^(1/2) + n^(1/3) + ... + n^(1/r) for abs(n) > 0 and r=floor(log_2(n)).

%H Paolo Xausa, <a href="/A089363/b089363.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A089361(10^n) = Sum_{p >= 2} (floor(10^(n/p)) - 1). - _David Wasserman_, Sep 14 2005

%e There are 16 perfect powers <= 100: 2^2, 2^3, 3^2, 2^4, 4^2, 5^2, 3^3, 2^5, 6^2, 7^2, 2^6, 4^3, 8^2, 3^4, 9^2, 10^2. So a(2) = 16.

%t A089363[n_] := Sum[Floor[10^(n/j)] - 1, {j, 2, BitLength[10^n] - 1}];

%t Array[A089363, 30] (* _Paolo Xausa_, Jan 14 2025 *)

%o (PARI) plessn10(n,m=2) = { for(k=1,n, s=0; z = 10^k; r = sqrtint(z); for(x=m,r, for(y=2,r, p = floor(x^y); if(p<=z,s++) ) ); print1(s", ") ) }

%Y Cf. A089361.

%K nonn

%O 1,1

%A _Cino Hilliard_, Dec 27 2003

%E More terms from _David Wasserman_, Sep 14 2005