A072776 Exponents of powers of squarefree numbers.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 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, 2, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Reinhard Zumkeller, Jul 10 2002

Keywords

• nonn

Comments

A072774(n) = A072775(n)^a(n);

A072774(n) is squarefree iff a(n)=1.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052409.

Programs

Haskell

a072776 n = a072776_list !! (n-1)  -- a072776_list defined in A072774.
-- Reinhard Zumkeller, Apr 06 2014

Python

from math import isqrt
from sympy import mobius, integer_nthroot, perfect_power
def A072776(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
    def f(x): return n-2+x-sum(g(integer_nthroot(x,k)[0]) for k in range(1,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 1 if not (p:=perfect_power(kmax)) else p[1] # Chai Wah Wu, Aug 19 2024