A084371 Squarefree kernels of powerful numbers (A001694).

1, 2, 2, 3, 2, 5, 3, 2, 6, 7, 2, 6, 3, 10, 6, 11, 5, 2, 6, 13, 14, 10, 6, 15, 3, 2, 6, 17, 6, 7, 19, 14, 10, 6, 21, 22, 10, 2, 23, 6, 5, 6, 15, 26, 3, 14, 10, 29, 6, 30, 31, 22, 6, 10, 2, 33, 15, 6, 34, 35, 6, 21, 11, 26, 37, 14, 38, 39, 14, 10, 41, 6, 42, 30, 43, 22, 6, 10

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Remark 2.5, p. 729.

Formula

a(n) = A007947(A001694(n)).

From Amiram Eldar, May 13 2023: (Start)

Sum_{A001694(k) < x} a(k) = (1/2) * x + o(x) (Jakimczuk, 2017). [corrected Sep 21 2024]

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/zeta(3/2))^2/2 = 0.1058641473... . (End)

Example

A001694(11) = 64 = 2^6 -> a(11) = 2,
A001694(12) = 72 = 2^3 * 3^2 -> a(12) = 2*3 = 6,
A001694(13) = 81 = 3^4 -> a(13) = 3.

Mathematica

s = {1}; Do[f = FactorInteger[n]; If[Min @ f[[;; , 2]] > 1, AppendTo[s, Times @@ f[[;; , 1]]]], {n, 2, 10^4}]; s (* Amiram Eldar, Aug 22 2019 *)

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = apply(x->rad(x), select(x->ispowerful(x), [1..nn])); \\ Michel Marcus, Aug 22 2019
(Python)
from math import prod, isqrt
from sympy import mobius, integer_nthroot, primefactors
def A084371(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n+x-squarefreepi(integer_nthroot(x, 3)[0]), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return prod(primefactors(bisection(f, n, n))) # Chai Wah Wu, Sep 13 2024

Crossrefs

Cf. A001694, A007947, A090699.

Sequence in context: A076403 A157987 A025478 * A025476 A347731 A331532

Adjacent sequences: A084368 A084369 A084370 * A084372 A084373 A084374

Keyword

nonn

Author

Reinhard Zumkeller, Jun 23 2003

Status

approved