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%I #10 Jan 29 2023 12:06:32
%S 1,1,1,1,1,1,1,1,10,1,1,1,1,1,1,1,1,10,1,1,1,1,1,1,26,1,1,1,1,1,1,1,1,
%T 1,1,10,1,1,1,1,1,1,1,1,10,1,1,1,50,26,1,1,1,1,1,1,1,1,1,1,1,1,10,1,1,
%U 1,1,1,1,1,1,10,1,1,26,1,1,1,1,1,82,1,1,1
%N a(n) is the sum of unitary divisors of n that are odd squares.
%C The unitary analog of A360159.
%H Amiram Eldar, <a href="/A360160/b360160.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} d.
%F a(n) = A358347(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269), and otherwise it equals A358347(n)/(A006519(n)+1).
%F Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = p^e + 1 if e is even and 1 if e is odd.
%F Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+2))/(2^(3*s)-4).
%F Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (2*sqrt(2)/(4*sqrt(2)-1)) * zeta(3/2)/(3*zeta(5/2)) = 0.3942576405... .
%t f[p_, e_] := If[OddQ[e], 1, p^e + 1]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1, if(f[i, 2]%2, 1, f[i, 1]^f[i, 2] + 1))); }
%Y Cf. A006519, A016754, A078434, A108269, A247041, A358347, A360159.
%K nonn,easy,mult
%O 1,9
%A _Amiram Eldar_, Jan 29 2023
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