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A365479 The sum of unitary divisors of the smallest square divisible by n.

%I A365479 #12 Sep 06 2023 01:10:03
%S A365479 1,5,10,5,26,50,50,17,10,130,122,50,170,250,260,17,290,50,362,130,500,
%T A365479 610,530,170,26,850,82,250,842,1300,962,65,1220,1450,1300,50,1370,
%U A365479 1810,1700,442,1682,2500,1850,610,260,2650,2210,170,50,130,2900,850,2810,410
%N A365479 The sum of unitary divisors of the smallest square divisible by n.
%C A365479 The number of unitary divisors of the smallest square divisible by n is the same as the number of unitary divisors of n, A034444(n).
%H A365479 Amiram Eldar, <a href="/A365479/b365479.txt">Table of n, a(n) for n = 1..10000</a>
%F A365479 a(n) = A034448(A053143(n)).
%F A365479 Multiplicative with a(p^e) = p^(e + (e mod 2)) + 1.
%F A365479 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-2) - 1/p^(3*s-2)).
%F A365479 Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 - 1/p^4 + 1/p^5 - 1/p^6) = 0.248414056414... .
%t A365479 f[p_, e_] := 1 + p^(e + Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
%o A365479 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^(f[i,2] + f[i,2]%2) + 1);}
%o A365479 (Python)
%o A365479 from math import prod
%o A365479 from sympy import factorint
%o A365479 def A365479(n): return prod(p**(e+(e&1))+1 for p,e in factorint(n).items()) # _Chai Wah Wu_, Sep 05 2023
%Y A365479 Cf. A002117, A034444, A034448, A053143, A365346, A365480, A365481.
%K A365479 nonn,easy,mult
%O A365479 1,2
%A A365479 _Amiram Eldar_, Sep 05 2023