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

1,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Refactorable Number.

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.

Example

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.

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 *)

Prog

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

Crossrefs

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

Partial sums of A336040.

Sequence in context: A109701 A124751 A103374 * A241087 A137722 A081305

Adjacent sequences: A208248 A208249 A208250 * A208252 A208253 A208254

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 12 2013

Status

approved