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A140697
Mobius transform of A000082.
1
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
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
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