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A365345 The number of divisors of the smallest square divisible by n.

%I A365345 #15 Apr 20 2025 08:52:27
%S A365345 1,3,3,3,3,9,3,5,3,9,3,9,3,9,9,5,3,9,3,9,9,9,3,15,3,9,5,9,3,27,3,7,9,
%T A365345 9,9,9,3,9,9,15,3,27,3,9,9,9,3,15,3,9,9,9,3,15,9,15,9,9,3,27,3,9,9,7,
%U A365345 9,27,3,9,9,27,3,15,3,9,9,9,9,27,3,15,5,9,3
%N A365345 The number of divisors of the smallest square divisible by n.
%C A365345 The sum of these divisors is A365346(n).
%C A365345 The number of divisors of the square root of the smallest square divisible by n is A322483(n).
%H A365345 Amiram Eldar, <a href="/A365345/b365345.txt">Table of n, a(n) for n = 1..10000</a>
%H A365345 Vaclav Kotesovec, <a href="/A365345/a365345.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A365345 a(n) = A000005(A053143(n)).
%F A365345 Multiplicative with a(p^e) = e + 1 + (e mod 2).
%F A365345 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 2/p^s - 1/p^(2*s)).
%F A365345 From _Vaclav Kotesovec_, Sep 05 2023: (Start)
%F A365345 Dirichlet g.f.: zeta(s)^3 * zeta(2*s) * Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
%F A365345 Let f(s) = Product_{primes p} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
%F A365345 Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n)^2/2 + (3*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)) * log(n) + 1 - 3*gamma + 3*gamma^2 - 3*sg1 + (3*gamma - 1)*12*zeta'(2)/Pi^2 + 12*zeta''(2)/Pi^2 + (12*zeta'(2)/Pi^2 + 3*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where
%F A365345 f(1) = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
%F A365345 f'(1) = f(1) * Sum_{primes p} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = 0.7343690473711153863995729489689746152413988981744946512300478410459132782...
%F A365345 f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} 4*p*(-1 + 2*p + p^2 - 4*p^3) * log(p)^2 / (1 - 3*p + p^2 + p^3)^2 = 0.1829055032494906699795154632343894745397324334876662084674149254022564139...,
%F A365345 gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
%t A365345 f[p_, e_] := e + 1 + Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365345 (PARI) a(n) = vecprod(apply(x -> x + 1 + x%2, factor(n)[, 2]));
%o A365345 (PARI) a(n) = numdiv(n*core(n)); \\ _Michel Marcus_, Sep 02 2023
%Y A365345 Cf. A000005, A053143, A322483, A365346, A365489, A365492.
%K A365345 nonn,easy,mult
%O A365345 1,2
%A A365345 _Amiram Eldar_, Sep 02 2023