A330018 - OEIS

%I #10 Jun 13 2020 00:50:13

%S 0,0,0,1,0,0,0,3,1,0,0,2,0,0,0,6,0,2,0,2,0,0,0,6,1,0,3,2,0,0,0,10,0,0,

%T 0,6,0,0,0,6,0,0,0,2,2,0,0,12,1,2,0,2,0,6,0,6,0,0,0,4,0,0,2,15,0,0,0,

%U 2,0,0,0,13,0,0,2,2,0,0,0,12,6,0,0,4,0,0,0,6,0,4

%N a(n) = Sum_{d|n} (bigomega(d) - omega(d)).

%C Inverse Moebius transform of A046660.

%H Robert Israel, <a href="/A330018/b330018.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} A046660(k) * x^k / (1 - x^k).

%F a(n) = A069264(n) - A062799(n).

%F If m and n are coprime, a(m*n) = tau(m)*a(n) + tau(n)*a(m), where tau = A000005. - _Robert Israel_, Jun 12 2020

%p N:= 100: # for a(1)..a(N)

%p V:= Vector(N):

%p for d from 1 to N do

%p   v:= add(t[2]-1, t=ifactors(d)[2]);

%p   L:= [seq(i,i=d..N,d)]:

%p   V[L]:= map(`+`,V[L],v);

%p od:

%p convert(V,list); # _Robert Israel_, Jun 12 2020

%t a[n_] := Sum[PrimeOmega[d] - PrimeNu[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 90}]

%o (PARI) a(n) = sumdiv(n, d, bigomega(d) - omega(d)); \\ _Michel Marcus_, Jun 12 2020

%Y Cf. A001221, A001222, A005117 (positions of 0's), A046660, A062799, A069264, A268340.

%K nonn

%O 1,8

%A _Ilya Gutkovskiy_, Nov 27 2019