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URL: https://oeis.org/A262209

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A262209
Inverse Möbius transform of A002654.
2
1, 2, 1, 3, 3, 2, 1, 4, 2, 6, 1, 3, 3, 2, 3, 5, 3, 4, 1, 9, 1, 2, 1, 4, 6, 6, 2, 3, 3, 6, 1, 6, 1, 6, 3, 6, 3, 2, 3, 12, 3, 2, 1, 3, 6, 2, 1, 5, 2, 12, 3, 9, 3, 4, 3, 4, 1, 6, 1, 9, 3, 2, 2, 7, 9, 2, 1, 9, 1, 6, 1, 8, 3, 6, 6, 3, 1, 6, 1, 15, 3, 6, 1, 3, 9
OFFSET
1,2
LINKS
FORMULA
G.f.: Sum_{k>=1} tau(k)*x^k/(1 + x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 14 2019
From Amiram Eldar, Feb 01 2025: (Start)
a(n) = Sum_{d|n} A002654(d).
Multiplicative with a(p^e) = e+1 if p = 2, a(p^e) = floor(e/2) + 1 if p == 3 (mod 4) (A002145), and a(p^e) = (e+1)*(e+2)/2 if p == 1 (mod 4) (A002144). (End)
MATHEMATICA
f[2, e_] := e + 1; f[p_, e_] := If[Mod[p, 4] == 1, (e + 1)*(e + 2)/2, Floor[e/2] + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, f[i, 2] + 1, if(f[i, 1] % 4 == 1, (f[i, 2]+1)*(f[i, 2]+2)/2, f[i, 2]\2 + 1))); } \\ Amiram Eldar, Feb 01 2025
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
R. J. Mathar, Sep 15 2015
STATUS
approved