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

1,2

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

Prog

(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

Crossrefs

Cf. A052410.

Sequence in context: A052410 A327501 A175781 * A304768 A243057 A090078

Adjacent sequences: A072772 A072773 A072774 * A072776 A072777 A072778

Keyword

nonn

Author

Reinhard Zumkeller, Jul 10 2002

Status

approved