A365680 - OEIS

%I #12 Sep 16 2023 02:21:29

%S 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,4,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,5,4,4,

%T 4,9,2,4,4,8,2,8,2,6,6,4,2,8,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,6,4,8,2,6,

%U 4,8,2,12,2,4,6,6,4,8,2,8,4,4,2,12,4,4

%N The number of exponentially squarefree divisors of n.

%C First differs from A252505 at n = 32.

%C The number of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.

%C The sum of these divisors is A365682(n) and the largest of them is A365683(n).

%H Amiram Eldar, <a href="/A365680/b365680.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = A013928(e+1) + 1.

%F a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

%t f[p_, e_] := Count[Range[e], _?SquareFreeQ] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) s(n) = sum(k=1, n, issquarefree(k)) + 1;

%o a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

%Y Cf. A000005, A013928, A046100, A209061, A252505, A365682, A365683.

%Y Similar sequences: A325837, A353898.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Sep 15 2023