A253642 Number of ways the perfect power A001597(n) can be written as a^b, with a, b > 1.

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

M. F. Hasler, Jan 25 2015

Keywords

• nonn

Comments

Run lengths of A072103. Also, the terms a(n) which exceed 1 constitute A175066. - Andrey Zabolotskiy, Aug 17 2016

Examples

			a(1)=0 since A001597(1)=1 can be written as a^b for a=1 and any b, but not using a base a > 1.
a(2)=a(3)=a(4)=1 since the following terms 4=2^2, 8=2^3 and 9=3^2 can be written as perfect powers in only one way.
a(5)=2 since A001597(5)=16=a^b for (a,b)=(2,4) and (4,2).
		

Crossrefs

Cf. A001597, A072103, A175064, A253641.

Programs

PARI

for(n=1,9999,(e=ispower(n))&&print1(numdiv(e)-1,","))

Python

from math import gcd
from sympy import mobius, integer_nthroot, divisor_count, factorint
def A253642(n):
    if n == 1: return 0
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return divisor_count(gcd(*factorint(kmax).values()))-1 # Chai Wah Wu, Aug 13 2024

Formula

a(n) = A000005(A253641(A001597(n))) - 1.

a(n) = A175064(n) - 1.