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A386470 The number of divisors of n whose exponents in their prime factorization are squares.

%I A386470 #8 Jul 28 2025 00:54:37
%S A386470 1,2,2,2,2,4,2,2,2,4,2,4,2,4,4,3,2,4,2,4,4,4,2,4,2,4,2,4,2,8,2,3,4,4,
%T A386470 4,4,2,4,4,4,2,8,2,4,4,4,2,6,2,4,4,4,2,4,4,4,4,4,2,8,2,4,4,3,4,8,2,4,
%U A386470 4,8,2,4,2,4,4,4,4,8,2,6,3,4,2,8,4,4,4
%N A386470 The number of divisors of n whose exponents in their prime factorization are squares.
%C A386470 First differs from A365171 and A369310 at n = 32.
%C A386470 First differs from A365488 at n = 128.
%C A386470 The number of terms in A197680 that divide n.
%C A386470 The sum of these divisors is A386471(n) and the largest of them is A386469(n).
%H A386470 Amiram Eldar, <a href="/A386470/b386470.txt">Table of n, a(n) for n = 1..10000</a>
%F A386470 Multiplicative with a(p^e) = A048760(e) + 1.
%F A386470 a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
%t A386470 f[p_, e_] := Floor[Sqrt[e]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A386470 (PARI) a(n) = vecprod(apply(x -> sqrtint(x) + 1, factor(n)[, 2]));
%Y A386470 Cf. A000005, A005117, A197680, A365171, A365488, A369310, A386469, A386471.
%K A386470 nonn,easy,mult
%O A386470 1,2
%A A386470 _Amiram Eldar_, Jul 22 2025