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A269131 Composite numbers whose largest prime factor is less than its second-largest prime factor's square, counting with multiplicity so that the factors of 18 are 2, 3, 3.

%I A269131 #37 Feb 13 2025 17:14:59
%S A269131 4,6,8,9,12,15,16,18,21,24,25,27,30,32,35,36,42,45,48,49,50,54,55,60,
%T A269131 63,64,65,70,72,75,77,81,84,85,90,91,95,96,98,100,105,108,110,115,119,
%U A269131 120,121,125,126,128,130,133,135,140,143,144,147,150,154,161,162,165
%N A269131 Composite numbers whose largest prime factor is less than its second-largest prime factor's square, counting with multiplicity so that the factors of 18 are 2, 3, 3.
%C A269131 These are numbers that a naïve factoring algorithm can declare done at the penultimate prime factor.
%H A269131 Marc Moskowitz, <a href="/A269131/b269131.txt">Table of n, a(n) for n = 1..10000</a>
%t A269131 cnQ[n_]:=Module[{pfs=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]}, CompositeQ[ n]&&Last[pfs]<pfs[[-2]]^2]; Select[Range[200],cnQ] (* _Harvey P. Dale_, Nov 05 2017 *)
%o A269131 (PARI) is(n)=my(f=factor(n),e=#f~); e && (f[e,2]>1 || (e>1 && f[e-1,1]^2>f[e,1])) \\ _Charles R Greathouse IV_, Feb 19 2016
%o A269131 (Python)
%o A269131 seq = []
%o A269131 for n in range(2, 1000):
%o A269131     temp = n
%o A269131     factor = 2
%o A269131     while temp > 1:
%o A269131         if temp % factor == 0:
%o A269131             temp //= factor
%o A269131             if temp == 1:
%o A269131                 continue
%o A269131             if temp < factor * factor:
%o A269131                 seq.append(n)
%o A269131         else:
%o A269131             factor += 1
%o A269131 print(seq)
%Y A269131 Cf. A002808, A251728 (semiprime terms).
%Y A269131 Subsequence of A253567.
%K A269131 nonn,easy
%O A269131 1,1
%A A269131 _Marc Moskowitz_, Feb 19 2016