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A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

%I A365481 #13 Sep 05 2023 19:13:24
%S A365481 1,3,4,3,6,12,8,5,4,18,12,12,14,24,24,5,18,12,20,18,32,36,24,20,6,42,
%T A365481 10,24,30,72,32,9,48,54,48,12,38,60,56,30,42,96,44,36,24,72,48,20,8,
%U A365481 18,72,42,54,30,72,40,80,90,60,72,62,96,32,9,84,144,68,54,96
%N A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.
%C A365481 The number of unitary divisors of the smallest number whose square is divisible by n is the same as the number of unitary divisors of n, A034444(n).
%H A365481 Amiram Eldar, <a href="/A365481/b365481.txt">Table of n, a(n) for n = 1..10000</a>
%F A365481 a(n) = A034448(A019554(n)).
%F A365481 Multiplicative with a(p^e) = p^(ceiling(e/2)) + 1.
%F A365481 Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-1) - 1/p^(3*s-1)).
%F A365481 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .
%t A365481 f[p_, e_] := p^Ceiling[e/2] + 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
%o A365481 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^ceil(f[i,2]/2) + 1);}
%o A365481 (Python)
%o A365481 from math import prod
%o A365481 from sympy import factorint
%o A365481 def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p,e in factorint(n).items()) # _Chai Wah Wu_, Sep 05 2023
%Y A365481 Cf. A002117, A013661, A019554, A034444, A034448, A365347, A365479, A365480.
%K A365481 nonn,easy,mult
%O A365481 1,2
%A A365481 _Amiram Eldar_, Sep 05 2023