A089363 Numbers of pairs (i, j), i, j > 1, such that i^j <= 10^n.

3, 16, 50, 145, 407, 1177, 3508, 10677, 32967, 102719, 321798, 1011538, 3186390, 10050746, 31730137, 100228044, 316713624, 1001037551, 3164497350, 10004755379, 31632975601, 100021893197, 316274794667, 1000101078155, 3162495003354, 10000467510250, 31623782520067

Offset: 1

Author

Cino Hilliard, Dec 27 2003

Keywords

• nonn

Comments

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)).

Examples

			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.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A089361.

Programs

Mathematica

A089363[n_] := Sum[Floor[10^(n/j)] - 1, {j, 2, BitLength[10^n] - 1}];
Array[A089363, 30] (* Paolo Xausa, Jan 14 2025 *)

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", ") ) }

Formula

a(n) = A089361(10^n) = Sum_{p >= 2} (floor(10^(n/p)) - 1). - David Wasserman, Sep 14 2005

Extensions

More terms from David Wasserman, Sep 14 2005