A384064 a(n) = s(n) divided by the smallest multiple prime factor of s(n), where s = A013929.

2, 4, 3, 6, 8, 6, 10, 12, 5, 9, 14, 16, 18, 20, 22, 15, 24, 7, 10, 26, 18, 28, 30, 21, 32, 34, 36, 15, 38, 40, 27, 42, 44, 30, 46, 48, 14, 33, 50, 52, 54, 56, 58, 39, 60, 11, 62, 25, 42, 64, 66, 45, 68, 70, 72, 21, 74, 30, 76, 51, 78, 80, 54, 82, 84, 13, 57, 86

Offset: 1

Author

Michael De Vlieger, Jun 23 2025

Keywords

• nonn
• easy

Comments

a(n) is the largest proper nonunitary divisor of s(n).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple additionally signifies powerful numbers that are not prime powers.

Crossrefs

Cf. A013929, A046027, A048105.

Programs

Mathematica

Map[#/Select[FactorInteger[#], #[[-1]] > 1 &, 1][[1, 1]] &, Select[Range[200], Not @* SquareFreeQ] ] (* or *) Map[Select[Most@ Divisors[#], Not @* CoprimeQ] &, Select[Range[200], Not @* SquareFreeQ] ][[All, -1]]

Python

from math import isqrt
from sympy import mobius, factorint
def A384064(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 m//next(p for p in sorted(s) if s[p]>1) # Chai Wah Wu, Jun 25 2025

Formula

a(n) = A013929(n)/A046027(n).