cp's OEIS Frontend

A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

%I A365208 #8 Aug 27 2023 01:46:40
%S A365208 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,2,4,4,6,2,8,2,6,4,4,
%T A365208 4,9,2,4,4,8,2,8,2,6,6,4,2,10,2,4,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,
%U A365208 6,4,8,2,12,2,4,4,6,4,8,2,10,5,4,2,12,4,4
%N A365208 The number of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).
%C A365208 First differs from A000005 at n = 25.
%C A365208 The sum of these divisors is A365209(n).
%H A365208 Amiram Eldar, <a href="/A365208/b365208.txt">Table of n, a(n) for n = 1..10000</a>
%F A365208 Multiplicative with a(p^e) = e+1 if p = 2 or 3, and a(p^e) = 2 for a prime p >= 5.
%F A365208 a(n) <= A000005(n), with equality if and only if n is not divisible by a square of a prime >= 5.
%F A365208 a(n) >= A034444(n), with equality if and only if n is neither divisible by 4 nor by 9.
%F A365208 a(n) = A000005(A065331(n)) * A034444(A065330(n)).
%F A365208 Dirichlet g.f.: (4^s/(4^s-1)) * (9^s/(9^s-1)) * zeta(s)^2/zeta(2*s).
%F A365208 Sum_{k==1..n} a(k) ~ (9/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - log(3)/4 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).
%t A365208 f[p_, e_] := If[p <= 3, e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365208 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, f[i,2]+1, 2));}
%Y A365208 Cf. A000005, A003586, A034444, A065330, A065331, A365209.
%Y A365208 Cf. A001620, A013661, A073002.
%K A365208 nonn,easy,mult
%O A365208 1,2
%A A365208 _Amiram Eldar_, Aug 26 2023