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
Examples
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,...).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
Programs
-
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 *)
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
Extensions
Definition corrected by N. J. A. Sloane, Jul 28 2008
More terms from Alois P. Heinz, Aug 28 2008
Comments