A065745 Sum of squares and twice squares dividing n.
1, 3, 1, 7, 1, 3, 1, 15, 10, 3, 1, 7, 1, 3, 1, 31, 1, 30, 1, 7, 1, 3, 1, 15, 26, 3, 10, 7, 1, 3, 1, 63, 1, 3, 1, 70, 1, 3, 1, 15, 1, 3, 1, 7, 10, 3, 1, 31, 50, 78, 1, 7, 1, 30, 1, 15, 1, 3, 1, 7, 1, 3, 10, 127, 1, 3, 1, 7, 1, 3, 1, 150, 1, 3, 26, 7, 1, 3, 1, 31, 91, 3, 1, 7, 1, 3, 1, 15, 1, 30, 1
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
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Mathematica
f[2, e_] := 2^(e+1) - 1; f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p^2-1), (p^(e+2)-1)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 13 2020 *)
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PARI
a(n) = sumdiv(n, d, issquare(d)*d + (1 - d%2)*issquare(d/2)*d) \\ Michel Marcus, Jun 17 2013
Formula
Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = (p^(e+2)-1)/(p-1)/(p+1) for odd p and even e and a(p^e) = (p^(e+1)-1)/(p-1)/(p+1) for odd p and odd e.
From Amiram Eldar, Dec 15 2023: (Start)
Dirichlet g.f.: (1 + 1/2^(s-1)) * zeta(2*s-2) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = ((2+sqrt(2))/6) * zeta(3/2) = 1.4865345575818562471... . (End)
Extensions
More terms from Matthew Conroy, Jan 19 2002