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A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).

%I A365209 #8 Aug 27 2023 01:46:07
%S A365209 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,26,
%T A365209 42,40,56,30,72,32,63,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U A365209 124,50,78,72,98,54,120,72,120,80,90,60,168,62,96,104,127,84,144
%N A365209 The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).
%C A365209 First differs from A000005 at n = 25.
%C A365209 The number of these divisors is A365208(n).
%H A365209 Amiram Eldar, <a href="/A365209/b365209.txt">Table of n, a(n) for n = 1..10000</a>
%F A365209 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.
%F A365209 a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.
%F A365209 a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.
%F A365209 a(n) = A000203(A065331(n)) * A034448(A065330(n)).
%F A365209 Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).
%F A365209 Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .
%t A365209 f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365209 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1), 1 + f[i,1]^f[i,2]));}
%Y A365209 Cf. A000203, A003586, A034448, A065330, A065331, A365208.
%Y A365209 Cf. A002117, A013661, A306633.
%K A365209 nonn,easy,mult
%O A365209 1,2
%A A365209 _Amiram Eldar_, Aug 26 2023