A306458 a(n) = A001694(n)/A007947(A001694(n)), the powerful numbers divided by their squarefree kernel.

1, 2, 4, 3, 8, 5, 9, 16, 6, 7, 32, 12, 27, 10, 18, 11, 25, 64, 24, 13, 14, 20, 36, 15, 81, 128, 48, 17, 54, 49, 19, 28, 40, 72, 21, 22, 50, 256, 23, 96, 125, 108, 45, 26, 243, 56, 80, 29, 144, 30, 31, 44, 162, 100, 512, 33, 75, 192, 34, 35, 216, 63, 121, 52

Offset

1,2

Comments

A permutation of the positive integers.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

A064549(a(n)) = A001694(n).

Maple

N:= 10^4: # to get terms corresponding to powerful numbers <= N
rad:= n -> convert(numtheory:-factorset(n), `*`):
S:= {1}:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  S:= S union map(t -> seq(t*p^i, i=2..floor(log[p](N/t))), select(`<=`, S, N/p^2));
od:
map(t -> t/rad(t), sort(convert(S, list))); # Robert Israel, Mar 20 2019

Mathematica

p=Join[{1}, Select[ Range@ 12500, Min@ FactorInteger[#][[All, 2]] > 1 &]]; rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);  p/(rad/@p) (* after Harvey P. Dale at A001694 and Robert G. Wilson v at A007947 *)

Prog

(PARI) apply(x->(x/factorback(factorint(x)[, 1])), select(x->ispowerful(x), vector(1600, k, k))) \\ Michel Marcus, Feb 17 2019
(Python)
from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A306458(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 = n+x-squarefreepi(integer_nthroot(x, 3)[0]), 0
        j = 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 (m:=bisection(f, n, n))//prod(primefactors(m)) # Chai Wah Wu, Sep 14 2024

Crossrefs

Cf. A001694, A007947, A064549.

Sequence in context: A075378 A101905 A297075 * A287637 A053211 A131390

Adjacent sequences: A306455 A306456 A306457 * A306459 A306460 A306461

Keyword

nonn,look

Author

Amiram Eldar, Feb 17 2019

Status

approved