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A369718
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The sum of unitary divisors of the smallest powerful number that is a multiple of n.
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4
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1, 5, 10, 5, 26, 50, 50, 9, 10, 130, 122, 50, 170, 250, 260, 17, 290, 50, 362, 130, 500, 610, 530, 90, 26, 850, 28, 250, 842, 1300, 962, 33, 1220, 1450, 1300, 50, 1370, 1810, 1700, 234, 1682, 2500, 1850, 610, 260, 2650, 2210, 170, 50, 130, 2900, 850, 2810, 140
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p) = p^2 + 1 and a(p^e) = p^e + 1 for e >= 2.
a(n) >= A034448(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(s-1) - 1/p^(2*s-3) + 1/p^(3*s-3) - 1/p^(3*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^4 + 1/p^6 - 2/p^7 + 1/p^8) = 0.73644353930922037459... .
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MATHEMATICA
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f[p_, e_] := If[e == 1, p^2 + 1, p^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, 2] == 1, 1 + f[i, 1]^2, 1 + f[i, 1]^f[i, 2])); }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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