A258567 - OEIS

%I #22 Sep 11 2024 00:32:40

%S 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 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 37,2,2,3,2,2,41,2,2,2,43,2,2,2,3,2,2,3

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

%H Reinhard Zumkeller, <a href="/A258567/b258567.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%o (Haskell)

%o a258567 = a020639 . a001694

%o (Python)

%o from math import isqrt

%o from sympy import mobius, integer_nthroot, primefactors

%o def A258567(n):

%o     def squarefreepi(n):

%o         return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))

%o     def bisection(f, kmin=0, kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x):

%o         c, l = n+x, 0

%o         j = isqrt(x)

%o         while j>1:

%o             k2 = integer_nthroot(x//j**2, 3)[0]+1

%o             w = squarefreepi(k2-1)

%o             c -= j*(w-l)

%o             l, j = w, isqrt(x//k2**3)

%o         c -= squarefreepi(integer_nthroot(x, 3)[0])-l

%o         return c

%o     return min(primefactors(bisection(f,n,n)),default=1) # _Chai Wah Wu_, Sep 10 2024

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

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jun 06 2015

%E Definition made more precise by _N. J. A. Sloane_, Apr 29 2024