A046027 Smallest multiple prime factor of the n-th nonsquarefree number (A013929).

2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, 2, 2, 3, 2, 7, 5, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 7, 3, 2, 2, 2, 2, 2, 3, 2, 11, 2, 5, 3, 2, 2, 3, 2, 2, 2, 7, 2, 5, 2, 3, 2, 2, 3, 2, 2, 13, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 11, 3, 2, 7, 2, 5, 2, 2, 2, 3

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Least Prime Factor.
• Eric Weisstein's World of Mathematics, Squareful.

Crossrefs

Cf. A013929, A020639, A046028, A249739.

Programs

Mathematica

Select[ FactorInteger[#], #[[2]]>1&, 1][[1, 1]]& /@ Select[ Range[300], !SquareFreeQ[#]& ] (* Jean-François Alcover, Nov 06 2012 *)

PARI

lista(nn) = apply(x->factor(x)[1,1], apply(x->x/core(x), select(x->!issquarefree(x), [1..nn]))); \\ Michel Marcus, Jun 24 2025

Python

from math import isqrt
from sympy import mobius, factorint
def A046027(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    s = factorint(m)
    return next(p for p in sorted(s) if s[p]>1) # Chai Wah Wu, Jul 22 2024

Formula

a(n) = A249739(A013929(n)). - Amiram Eldar, Feb 11 2021