A140697 Mobius transform of A000082.

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

Offset

1,2

Comments

Dirichlet convolution of the sequence of (absolute values of A055615) and A007434. - R. J. Mathar, Feb 27 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

Dirichlet g.f.: zeta(s-1)*zeta(s-2)/(zeta(2s-2)*zeta(s)). - R. J. Mathar, Feb 27 2011

Sum_{k=1..n} a(k) ~ 5*n^3 / (Pi^2 * zeta(3)). - Vaclav Kotesovec, Jan 11 2019

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

Example

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,...).

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A000082, A002117, A007434, A054525, A055615.

Sequence in context: A094684 A240438 A365417 * A048253 A102174 A140515

Adjacent sequences: A140694 A140695 A140696 * A140698 A140699 A140700

Keyword

nonn,easy,mult

Author

Gary W. Adamson, May 23 2008

Extensions

Definition corrected by N. J. A. Sloane, Jul 28 2008

More terms from Alois P. Heinz, Aug 28 2008

Status

approved