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A349692
Dirichlet convolution of the gcd-sum function (A018804) with itself.
1
1, 6, 10, 25, 18, 60, 26, 88, 67, 108, 42, 250, 50, 156, 180, 280, 66, 402, 74, 450, 260, 252, 90, 880, 211, 300, 372, 650, 114, 1080, 122, 832, 420, 396, 468, 1675, 146, 444, 500, 1584, 162, 1560, 170, 1050, 1206, 540, 186, 2800, 435, 1266, 660, 1250, 210, 2232, 756
OFFSET
1,2
FORMULA
Dirichlet g.f.: zeta(s-1)^4 / zeta(s)^2.
a(n) = Sum_{d|n} A018804(d) * A018804(n/d).
a(n) = Sum_{d|n} A029935(d) * A038040(n/d).
a(n) = Sum_{d|n} A344683(d) * A000203(n/d).
Multiplicative with a(p^e) = (e+1) * p^(e-2) * ((e+2)*(e+3)*p^2 - 2*e*(e+2)*p + e*(e-1))/6. - Amiram Eldar, Nov 25 2021
MATHEMATICA
A018804[n_] := Sum[GCD[n, k], {k, 1, n}]; a[n_] := Sum[A018804[d] A018804[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e + 1)*p^(e - 2)*((e + 2)*(e + 3)*p^2 - 2*e*(e + 2)*p + e*(e - 1))/6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 25 2021 *)
PROG
(PARI)
A029935(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ From A029935.
A349692(n) = sumdiv(n, d, A029935(n/d)*d*numdiv(d)); \\ Antti Karttunen, Nov 25 2021
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Nov 25 2021
STATUS
approved