A208251 Number of refactorable numbers less than or equal to n.

1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Wesley Ivan Hurt, Jan 12 2013

Keywords

• nonn
• easy

Comments

A number is refactorable if it is divisible by the number of its divisors.

Examples

			a(1) = 1 since 1 is the first refactorable number, a(2) = 2 since there are two refactorable numbers less than or equal to 2, a(3) through a(7) = 2 since the next refactorable number is 8.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Refactorable Number.

Crossrefs

Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.

Partial sums of A336040.

Programs

Maple

with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);

Mathematica

Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* Amiram Eldar, Oct 11 2023 *)

PARI

a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ Michel Marcus, Sep 10 2018

Formula

a(n) = Sum_{i=1..n} 1 + floor(i/d(i)) - ceiling(i/d(i)), where d(n) is the number of divisors of n.