A376833 - OEIS

%I #14 Oct 07 2024 00:43:31

%S 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 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 47,71,13,29,73,7,31,79,53,23,5,83,5,3,59,89,7,61,37,3

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

%H Michael De Vlieger, <a href="/A376833/b376833.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A376833/a376833.png">Log log scatterplot of a(n)</a>, n = 1..2^20.

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

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

%e Let b(n) = A120944(n).

%e a(1) = 3 since b(1) = 6, and 3 is the second smallest prime factor.

%e a(2) = 5 since b(2) = 10, and 5 is the second smallest prime factor.

%e Table showing select values of a(n):

%e     n   b(n)          a(n)

%e   -----------------------

%e    1    6 = 2*3        3

%e    2   10 = 2*5        5

%e    3   14 = 2*7        7

%e    4   15 = 3*5        5

%e    5   21 = 3*7        7

%e    6   22 = 2*11      11

%e    7   26 = 2*13      13

%e    8   30 = 2*3*5      3

%e   14   42 = 2*3*7      3

%e   22   66 = 2*3*11     3

%e   24   70 = 2*5*7      5

%e   82  210 = 2*3*5*7    3

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

%o (Python)

%o from math import isqrt

%o from sympy import primepi, mobius, primefactors

%o def A376833(n):

%o     def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o     m, k = n+1, f(n+1)

%o     while m != k:

%o         m, k = k, f(k)

%o     return primefactors(m)[1] # _Chai Wah Wu_, Oct 06 2024

%Y Cf. A119288, A120944.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Oct 05 2024