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A374538
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a(n) is the sum of the squares of the unitary divisors of n that are exponentially odd numbers (A268335).
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2
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1, 5, 10, 1, 26, 50, 50, 65, 1, 130, 122, 10, 170, 250, 260, 1, 290, 5, 362, 26, 500, 610, 530, 650, 1, 850, 730, 50, 842, 1300, 962, 1025, 1220, 1450, 1300, 1, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 122, 26, 2650, 2210, 10, 1, 5, 2900, 170, 2810, 3650, 3172
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OFFSET
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1,2
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COMMENTS
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The number of unitary divisors of n that are exponentially odd is A055076(n) and their sum is A358346(n).
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LINKS
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FORMULA
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a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A374537(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^(2*e) + 1 if e is odd, and 1 otherwise.
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.79482441214759383925... .
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MATHEMATICA
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f[p_, e_] := 1 + If[OddQ[e], p^(2*e), 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2, f[i, 1]^(2*f[i, 2]), 0)); }
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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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