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A140697
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Mobius transform of A000082.
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1
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1, 5, 11, 18, 29, 55, 55, 72, 96, 145, 131, 198, 181, 275, 319, 288, 305, 480, 379, 522, 605, 655, 551, 792, 720, 905, 864, 990, 869, 1595, 991, 1152, 1441, 1525, 1595, 1728, 1405, 1895, 1991, 2088, 1721, 3025, 1891, 2358, 2784, 2755, 2255, 3168, 2688, 3600
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s-1)*zeta(s-2)/(zeta(2s-2)*zeta(s)). - R. J. Mathar, Feb 27 2011
Multiplicative with a(p) = p*(p+1) - 1, and a(p^e) = (p-1)*(p+1)^2*p^(2*e-3) for e >= 2. - Amiram Eldar, Oct 28 2023
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EXAMPLE
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a(4) = 18 = (0, -1, 0, 1) dot (1, 6, 12, 24), where (0, -1 0, 1) = row 4 of A054525 and A000082 = (1, 6, 12, 24, 30, 72,...).
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MAPLE
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with (numtheory): a:= n-> add (k^2* mul(1+1/p, p=factorset(k)) *mobius (n/k), k=divisors(n)): seq (a(n), n=1..60); # Alois P. Heinz, Aug 28 2008
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MATHEMATICA
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a[n_] := Sum[ k^2*Product[ 1+1/p, {p, FactorInteger[k][[All, 1]]}]* MoebiusMu[n/k], {k, Divisors[n]}] - MoebiusMu[n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 03 2012, after Alois P. Heinz *)
f[p_, e_] := (p - 1)*(p + 1)^2*p^(2*e - 3); f[p_, 1] := p*(p + 1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)
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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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EXTENSIONS
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STATUS
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approved
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