A376833 Second smallest prime factor of numbers m that are both squarefree and composite.

3, 5, 7, 5, 7, 11, 13, 3, 11, 17, 7, 19, 13, 3, 23, 17, 11, 19, 29, 31, 13, 3, 23, 5, 37, 11, 3, 41, 17, 43, 29, 13, 31, 47, 19, 3, 5, 53, 5, 37, 3, 23, 59, 17, 61, 41, 43, 5, 19, 67, 3, 47, 71, 13, 29, 73, 7, 31, 79, 53, 23, 5, 83, 5, 3, 59, 89, 7, 61, 37, 3

Offset: 1

Author

Michael De Vlieger, Oct 05 2024

Keywords

• nonn
• easy

Examples

			Let b(n) = A120944(n).
a(1) = 3 since b(1) = 6, and 3 is the second smallest prime factor.
a(2) = 5 since b(2) = 10, and 5 is the second smallest prime factor.
Table showing select values of a(n):
    n   b(n)          a(n)
  -----------------------
   1    6 = 2*3        3
   2   10 = 2*5        5
   3   14 = 2*7        7
   4   15 = 3*5        5
   5   21 = 3*7        7
   6   22 = 2*11      11
   7   26 = 2*13      13
   8   30 = 2*3*5      3
  14   42 = 2*3*7      3
  22   66 = 2*3*11     3
  24   70 = 2*5*7      5
  82  210 = 2*3*5*7    3
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Crossrefs

Cf. A119288, A120944.

Programs

Mathematica

Map[FactorInteger[#][[2, 1]] &, Select[Range[250], And[SquareFreeQ[#], CompositeQ[#]] &]]

Python

from math import isqrt
from sympy import primepi, mobius, primefactors
def A376833(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return primefactors(m)[1] # Chai Wah Wu, Oct 06 2024

Formula

a(n) = A119288(A120944(n)).

For even squarefree semiprime A120944(n) = 2*p with odd prime p, a(n) = p sets a record in this sequence.