A305053 If n = Product_i prime(x_i)^k_i, then a(n) = Sum_i k_i * omega(x_i) - omega(n), where omega = A001221 is number of distinct prime factors.

0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 1, 0, 2, -1, 1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 1, -1, 1, -1, 1, -1, 0, -1, 0, -1, 1, 0, 1, 0, 1, -1, 0

Offset: 1

Author

Gus Wiseman, May 24 2018

Keywords

• sign

Examples

			2925 = prime(2)^2 * prime(3)^2 * prime(6)^1, so a(2925) = 2*1 + 2*1 + 1*2 - 3 = 3.
		

Crossrefs

Cf. A001221, A003963, A056239, A112798, A286520, A290103, A302242, A304714, A304716, A305052, A305054.

Programs

Mathematica

Table[If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>(k*PrimeNu[PrimePi[p]]-1)]],{n,100}]

PARI

a(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,2]*omega(primepi(f[k,1]))) - omega(n);} \\ Michel Marcus, Jun 09 2018

Formula

Totally additive with a(prime(n)) = omega(n) - 1.

a(n) = A305054(n) - A001221(n). - Michel Marcus, Jun 09 2018