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A360160
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a(n) is the sum of unitary divisors of n that are odd squares.
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2
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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, 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, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 82, 1, 1, 1
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OFFSET
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1,9
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} d.
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.
Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+2))/(2^(3*s)-4).
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... .
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MATHEMATICA
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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]
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PROG
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(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))); }
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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