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%I #26 Oct 28 2023 04:08:50
%S 1,5,11,18,29,55,55,72,96,145,131,198,181,275,319,288,305,480,379,522,
%T 605,655,551,792,720,905,864,990,869,1595,991,1152,1441,1525,1595,
%U 1728,1405,1895,1991,2088,1721,3025,1891,2358,2784,2755,2255,3168,2688,3600
%N Mobius transform of A000082.
%C Dirichlet convolution of the sequence of (absolute values of A055615) and A007434. - _R. J. Mathar_, Feb 27 2011
%H Alois P. Heinz, <a href="/A140697/b140697.txt">Table of n, a(n) for n = 1..10000</a>
%F Dirichlet g.f.: zeta(s-1)*zeta(s-2)/(zeta(2s-2)*zeta(s)). - _R. J. Mathar_, Feb 27 2011
%F Sum_{k=1..n} a(k) ~ 5*n^3 / (Pi^2 * zeta(3)). - _Vaclav Kotesovec_, Jan 11 2019
%F 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
%e 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,...).
%p 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
%t 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_ *)
%t 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 *)
%Y Cf. A000082, A002117, A007434, A054525, A055615.
%K nonn,easy,mult
%O 1,2
%A _Gary W. Adamson_, May 23 2008
%E Definition corrected by _N. J. A. Sloane_, Jul 28 2008
%E More terms from _Alois P. Heinz_, Aug 28 2008
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