%I #14 Aug 31 2023 05:32:59
%S 1,1,3,3,7,3,11,7,12,7,19,9,23,11,21,16,31,12,35,21,33,19,43,21,48,23,
%T 44,33,55,21,59,36,57,31,77,36,71,35,69,49,79,33,83,57,84,43,91,48,
%U 108,48,93,69,103,44,133,77,105,55,115,63,119,59,132,80,161,57,131,93,129
%N A054525 * A029935.
%H Amiram Eldar, <a href="/A143275/b143275.txt">Table of n, a(n) for n = 1..10000</a>
%F Möbius transform (A054525) of A029935: (1, 2, 4, 5, 8, 8, 12, 12, 16, ...).
%F Multiplicative with a(p) = 2*p - 3 and a(p^e) = (e*(p-1) + p + 2) * (p-1)^2 * p^(e-3) for e > 1. - _Amiram Eldar_, Aug 31 2023
%e a(4) = 3 = (0, -1, 0, 1) dot (1, 2, 4, 5) = (0 - 2 + 0 + 5), where K(0, -1, 0, 1) = row 4 of A054525 and A143275 = (1, 2, 4, 5, 8, 8, 12, ...).
%p read("transforms") : A029935 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) do a := a+ numtheory[phi](d)*numtheory[phi](n/d); od; RETURN(a) ; end: a029935 := [seq(A029935(n),n=1..300)] ; a143275 := MOBIUS(a029935) ; # _R. J. Mathar_, Jan 19 2009
%t f[p_, e_] := If[e > 1, (e*(p-1) + p + 2) * (p-1)^2 * p^(e-3), 2*p - 3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 31 2023 *)
%Y Cf. A029935, A054525.
%K nonn,easy,mult
%O 1,3
%A _Gary W. Adamson_, Aug 03 2008
%E More terms from _R. J. Mathar_, Jan 19 2009