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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.
2
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
1,27
FORMULA
Totally additive with a(prime(n)) = omega(n) - 1.
a(n) = A305054(n) - A001221(n). - Michel Marcus, Jun 09 2018
EXAMPLE
2925 = prime(2)^2 * prime(3)^2 * prime(6)^1, so a(2925) = 2*1 + 2*1 + 1*2 - 3 = 3.
MATHEMATICA
Table[If[n==1, 0, Total@Cases[FactorInteger[n], {p_, k_}:>(k*PrimeNu[PrimePi[p]]-1)]], {n, 100}]
PROG
(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
KEYWORD
sign
AUTHOR
Gus Wiseman, May 24 2018
STATUS
approved