A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).

1, 2, 2, 3, 2, 5, 3, 2, 2, 7, 2, 2, 3, 2, 2, 11, 5, 2, 2, 13, 2, 2, 2, 3, 3, 2, 2, 17, 2, 7, 19, 2, 2, 2, 3, 2, 2, 2, 23, 2, 5, 2, 3, 2, 3, 2, 2, 29, 2, 2, 31, 2, 2, 2, 2, 3, 3, 2, 2, 5, 2, 3, 11, 2, 37, 2, 2, 3, 2, 2, 41, 2, 2, 2, 43, 2, 2, 2, 3, 2, 2, 3

Offset

1,2

Links

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

Formula

a(n) = A020639(A001694(n)).

a(A258599(n)) = A000040(n) and a(m) != A000040(n) for m < A258599(n).

Mathematica

Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 1 || n == 1, f[[1, 1]], Nothing], {n, 1, 3000}] (* Amiram Eldar, Jan 30 2023 *)

Prog

(Haskell)
a258567 = a020639 . a001694
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot, primefactors
def A258567(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, 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)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return min(primefactors(bisection(f, n, n)), default=1) # Chai Wah Wu, Sep 10 2024

Crossrefs

Cf. A001694, A020639, A258599, A258568, A258569, A258570, A258571.

Sequence in context: A237984 A118136 A356552 * A076396 A370834 A076397

Adjacent sequences: A258564 A258565 A258566 * A258568 A258569 A258570

Keyword

nonn

Author

Reinhard Zumkeller, Jun 06 2015

Extensions

Definition made more precise by N. J. A. Sloane, Apr 29 2024

Status

approved