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A144523 Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n > number of prime factors in n+1.

%I A144523 #13 Aug 23 2020 11:13:44
%S A144523 36,210,300,528,1035,1176,1275,1485,1596,2080,2346,2926,3240,3321,
%T A144523 3570,4095,4278,5460,5565,6105,6555,6903,7260,8256,8778,9870,10440,
%U A144523 11628,11935,12880,13695,14196,15576,16653,17020,17391,17955,20100,20910,21736,22578,23436,24310,25200,25425
%N A144523 Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n > number of prime factors in n+1.
%C A144523 Subsequence of A144291 - _R. J. Mathar_, Jan 17 2009
%C A144523 Prime factors counted with multiplicity. - _Harvey P. Dale_, Aug 23 2020
%H A144523 Harvey P. Dale, <a href="/A144523/b144523.txt">Table of n, a(n) for n = 1..1000</a>
%e A144523 If n=8=2*2*2(number of prime factors = 3) and n+1=9=3*3(number of prime factors = 2), then 8*9/2=36=a(1). If n=20=2*2*5(number of prime factors = 3) and n+1=21=3*7(number of prime factors = 2), then 20*21/2=210=a(2). If n=24=2*2*2*3(number of prime factors = 4) and n+1=25=5*5(number of prime factors = 2), then 24*25/2=300=a(3), etc.
%t A144523 (Times@@#)/2&/@Select[Partition[Range[250],2,1],AllTrue[ #,CompositeQ] && PrimeOmega[#[[1]]]>PrimeOmega[#[[2]]]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 23 2020 *)
%Y A144523 Cf. A000040, A000392, A141468, A152622.
%K A144523 nonn
%O A144523 1,1
%A A144523 _Juri-Stepan Gerasimov_, Dec 15 2008
%E A144523 Corrected definition. 2926 inserted and extended. - _R. J. Mathar_, Jan 17 2009