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A072079
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Sum of 3-smooth divisors of n.
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5
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1, 3, 4, 7, 1, 12, 1, 15, 13, 3, 1, 28, 1, 3, 4, 31, 1, 39, 1, 7, 4, 3, 1, 60, 1, 3, 40, 7, 1, 12, 1, 63, 4, 3, 1, 91, 1, 3, 4, 15, 1, 12, 1, 7, 13, 3, 1, 124, 1, 3, 4, 7, 1, 120, 1, 15, 4, 3, 1, 28, 1, 3, 13, 127, 1, 12, 1, 7, 4, 3, 1, 195, 1, 3, 4, 7, 1, 12, 1, 31, 121
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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(2^e) = 2^(e+1)-1, a(3^e) = (3^(e+1)-1)/2, a(p^e) = 1, p>3. Christian G. Bower, May 20 2005
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3)).
Sum_{k=1..n} a(k) ~ c_1 * (n * log(n)^2 + c_2 * n * log(n) + c_3 * n), where c_1 = 1/(2*log(2)*log(3)) = 0.656598..., c_2 = (2*gamma - 2 + log(6)) = 0.9461907..., and c_3 = (log(6)^2 + log(2)*log(3))/6 - (log(6)-2)*(1-gamma) - 2*gamma_1 = 0.895656..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633). (End)
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MATHEMATICA
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f[p_, e_] := If[p > 3, 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
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PROG
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(PARI) a(n) = (2^(valuation(n, 2)+1)-1)*(3^(valuation(n, 3)+1)-1)/2; \\ Amiram Eldar, Dec 01 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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