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

%I A258567 #22 Sep 11 2024 00:32:40
%S A258567 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,
%T A258567 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,
%U A258567 37,2,2,3,2,2,41,2,2,2,43,2,2,2,3,2,2,3
%N A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).
%H A258567 Reinhard Zumkeller, <a href="/A258567/b258567.txt">Table of n, a(n) for n = 1..10000</a>
%F A258567 a(n) = A020639(A001694(n)).
%F A258567 a(A258599(n)) = A000040(n) and a(m) != A000040(n) for m < A258599(n).
%t A258567 Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 1 || n == 1, f[[1, 1]], Nothing], {n, 1, 3000}] (* _Amiram Eldar_, Jan 30 2023 *)
%o A258567 (Haskell)
%o A258567 a258567 = a020639 . a001694
%o A258567 (Python)
%o A258567 from math import isqrt
%o A258567 from sympy import mobius, integer_nthroot, primefactors
%o A258567 def A258567(n):
%o A258567     def squarefreepi(n):
%o A258567         return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A258567     def bisection(f, kmin=0, kmax=1):
%o A258567         while f(kmax) > kmax: kmax <<= 1
%o A258567         while kmax-kmin > 1:
%o A258567             kmid = kmax+kmin>>1
%o A258567             if f(kmid) <= kmid:
%o A258567                 kmax = kmid
%o A258567             else:
%o A258567                 kmin = kmid
%o A258567         return kmax
%o A258567     def f(x):
%o A258567         c, l = n+x, 0
%o A258567         j = isqrt(x)
%o A258567         while j>1:
%o A258567             k2 = integer_nthroot(x//j**2, 3)[0]+1
%o A258567             w = squarefreepi(k2-1)
%o A258567             c -= j*(w-l)
%o A258567             l, j = w, isqrt(x//k2**3)
%o A258567         c -= squarefreepi(integer_nthroot(x, 3)[0])-l
%o A258567         return c
%o A258567     return min(primefactors(bisection(f,n,n)),default=1) # _Chai Wah Wu_, Sep 10 2024
%Y A258567 Cf. A001694, A020639, A258599, A258568, A258569, A258570, A258571.
%K A258567 nonn
%O A258567 1,2
%A A258567 _Reinhard Zumkeller_, Jun 06 2015
%E A258567 Definition made more precise by _N. J. A. Sloane_, Apr 29 2024