cp's OEIS Frontend

A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.

%I A360162 #9 Jan 29 2023 12:10:16
%S A360162 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,5,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,1,1,1,
%T A360162 1,12,1,1,1,1,1,1,1,3,4,1,1,5,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,9,1,1,1,3,
%U A360162 1,1,1,4,1,1,6,3,1,1,1,5,10,1,1,3,1,1,1
%N A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares.
%C A360162 The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).
%C A360162 The unitary analog of A069290.
%H A360162 Amiram Eldar, <a href="/A360162/b360162.txt">Table of n, a(n) for n = 1..10000</a>
%F A360162 a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).
%F A360162 Multiplicative with a(p^e) = p^(e/2) + 1 if e is even, and 1 if e is odd.
%F A360162 Dirichlet g.f.: zeta(s)*zeta(2*s-1)/zeta(3*s-1).
%F A360162 Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
%t A360162 f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360162 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1)); }
%Y A360162 Cf. A001620, A056624, A069290, A306016, A358347.
%K A360162 nonn,easy,mult
%O A360162 1,4
%A A360162 _Amiram Eldar_, Jan 29 2023