A072775 Squarefree kernels of powers of squarefree numbers.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 13, 14, 15, 2, 17, 19, 21, 22, 23, 5, 26, 3, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 41, 42, 43, 46, 47, 7, 51, 53, 55, 57, 58, 59, 61, 62, 2, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 3, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 10, 101

Offset: 1

Author

Reinhard Zumkeller, Jul 10 2002

Keywords

• nonn

Comments

a(n) = A007947(A072774(n));

A072774(n) = a(n)^A072776(n);

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

Links

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

Crossrefs

Cf. A052410.

Programs

Haskell

a072775 n = a072775_list !! (n-1)  -- a072775_list defined in A072774.
-- Reinhard Zumkeller, Apr 06 2014

Python

from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A072775(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 prod(primefactors(kmax)) # Chai Wah Wu, Aug 19 2024