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

1,4

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

Prog

(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

Crossrefs

Cf. A052409.

Sequence in context: A373835 A373369 A319864 * A077481 A278113 A242012

Adjacent sequences: A072773 A072774 A072775 * A072777 A072778 A072779

Keyword

nonn

Author

Reinhard Zumkeller, Jul 10 2002

Status

approved