A081400 a(n) = d(n) - bigomega(n) - A005361(n).

0, 0, 0, -1, 0, 1, 0, -2, -1, 1, 0, 1, 0, 1, 1, -3, 0, 1, 0, 1, 1, 1, 0, 1, -1, 1, -2, 1, 0, 4, 0, -4, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, -1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 6, 0, 1, 1, -5, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, -3, 1, 0, 6, 1, 1, 1, 1, 0, 6, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 4, 1, 0, 1, 0, 4, 1, 1, 0, 4, 1, 1, 1, 1, 1, 8, -1, 1

Offset: 1

Author

Labos Elemer, Mar 28 2003

Keywords

• sign

Examples

			Negative for true prime powers; zero for 1 and primes; see also A030231, A007304, A034683, A075819 etc. to judge about positivity or magnitude.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001222, A005361, A030231, A030230, A007304, A034683, A075819.

Programs

PARI

a(n) = my(f=factor(n)); numdiv(n) - bigomega(n) - prod(k=1, #f~, f[k,2]); \\ Michel Marcus, May 25 2017

Python

from sympy import primefactors, factorint, divisor_count
from operator import mul
def bigomega(n): return 0 if n==1 else bigomega(n/primefactors(n)[0]) + 1
def a005361(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [f[i] for i in f])
def a(n): return divisor_count(n) - bigomega(n) - a005361(n) # Indranil Ghosh, May 25 2017

Formula

a(n) = A000005(n) - A001222(n) - A005361(n).