A330018 a(n) = Sum_{d|n} (bigomega(d) - omega(d)).

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, 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, 2, 0, 0, 0, 13, 0, 0, 2, 2, 0, 0, 0, 12, 6, 0, 0, 4, 0, 0, 0, 6, 0, 4

Offset: 1

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

• nonn

Comments

Inverse Moebius transform of A046660.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N do
  v:= add(t[2]-1, t=ifactors(d)[2]);
  L:= [seq(i,i=d..N,d)]:
  V[L]:= map(`+`,V[L],v);
od:
convert(V,list); # Robert Israel, Jun 12 2020

Mathematica

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

PARI

a(n) = sumdiv(n, d, bigomega(d) - omega(d)); \\ Michel Marcus, Jun 12 2020

Formula

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

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

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