cp's OEIS Frontend

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

%I A208251 #32 Feb 16 2025 08:33:16
%S A208251 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,
%T A208251 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,
%U A208251 11,11,11,11,11,11
%N A208251 Number of refactorable numbers less than or equal to n.
%C A208251 A number is refactorable if it is divisible by the number of its divisors.
%H A208251 Robert Israel, <a href="/A208251/b208251.txt">Table of n, a(n) for n = 1..10000</a>
%H A208251 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RefactorableNumber.html">Refactorable Number</a>.
%F A208251 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.
%e A208251 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.
%p A208251 with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);
%t A208251 Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* _Amiram Eldar_, Oct 11 2023 *)
%o A208251 (PARI) a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ _Michel Marcus_, Sep 10 2018
%Y A208251 Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.
%Y A208251 Partial sums of A336040.
%K A208251 nonn,easy
%O A208251 1,2
%A A208251 _Wesley Ivan Hurt_, Jan 12 2013