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

%I A376833 #14 Oct 07 2024 00:43:31
%S A376833 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,
%T A376833 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,
%U A376833 47,71,13,29,73,7,31,79,53,23,5,83,5,3,59,89,7,61,37,3
%N A376833 Second smallest prime factor of numbers m that are both squarefree and composite.
%H A376833 Michael De Vlieger, <a href="/A376833/b376833.txt">Table of n, a(n) for n = 1..10000</a>
%H A376833 Michael De Vlieger, <a href="/A376833/a376833.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%F A376833 a(n) = A119288(A120944(n)).
%F A376833 For even squarefree semiprime A120944(n) = 2*p with odd prime p, a(n) = p sets a record in this sequence.
%e A376833 Let b(n) = A120944(n).
%e A376833 a(1) = 3 since b(1) = 6, and 3 is the second smallest prime factor.
%e A376833 a(2) = 5 since b(2) = 10, and 5 is the second smallest prime factor.
%e A376833 Table showing select values of a(n):
%e A376833     n   b(n)          a(n)
%e A376833   -----------------------
%e A376833    1    6 = 2*3        3
%e A376833    2   10 = 2*5        5
%e A376833    3   14 = 2*7        7
%e A376833    4   15 = 3*5        5
%e A376833    5   21 = 3*7        7
%e A376833    6   22 = 2*11      11
%e A376833    7   26 = 2*13      13
%e A376833    8   30 = 2*3*5      3
%e A376833   14   42 = 2*3*7      3
%e A376833   22   66 = 2*3*11     3
%e A376833   24   70 = 2*5*7      5
%e A376833   82  210 = 2*3*5*7    3
%t A376833 Map[FactorInteger[#][[2, 1]] &, Select[Range[250], And[SquareFreeQ[#], CompositeQ[#]] &]]
%o A376833 (Python)
%o A376833 from math import isqrt
%o A376833 from sympy import primepi, mobius, primefactors
%o A376833 def A376833(n):
%o A376833     def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A376833     m, k = n+1, f(n+1)
%o A376833     while m != k:
%o A376833         m, k = k, f(k)
%o A376833     return primefactors(m)[1] # _Chai Wah Wu_, Oct 06 2024
%Y A376833 Cf. A119288, A120944.
%K A376833 nonn,easy
%O A376833 1,1
%A A376833 _Michael De Vlieger_, Oct 05 2024