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A114436 Indices of 5-almost prime triangular numbers.

%I A114436 #18 Feb 16 2025 08:32:59
%S A114436 15,24,27,31,35,39,44,47,54,55,56,71,72,75,79,81,84,87,90,98,107,108,
%T A114436 112,116,124,132,134,140,147,153,155,162,164,167,170,171,174,179,180,
%U A114436 183,184,199,203,204,209,219,220,225,230,234,244,245,247,248,249
%N A114436 Indices of 5-almost prime triangular numbers.
%H A114436 Vincenzo Librandi, <a href="/A114436/b114436.txt">Table of n, a(n) for n = 1..1000</a>
%H A114436 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A114436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number.</a>
%H A114436 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A114436 {a(n)} = {k such that A001222(A000217(k)) = 5}. {a(n)} = {k such that k*(k+1)/2 has exactly 5 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014614}.
%F A114436 { m :  A069904(m) = 5 }. - _Alois P. Heinz_, Aug 05 2019
%e A114436 a(1) = 15 because T(15) = TriangularNumber(15) = 15*(15+1)/2 = 120 = 2^3 * 3 * 5 is a 5-almost prime.
%e A114436 a(2) = 24 because T(24) = 24*(24+1)/2 = 300 = 2^2 * 3 * 5^2 is a 5-almost prime.
%e A114436 a(3) = 27 because T(27) = 27*(27+1)/2 = 378 = 2 * 3^3 * 7 is a 5-almost prime.
%e A114436 a(4) = 31 because T(27) = 31*(31+1)/2 = 496 = 2^4 * 31 is a 5-almost prime.
%e A114436 a(17) = 84 because T(27) = 84*(84+1)/2 = 3570 = 2 * 3 * 5 * 7 * 17 is a 5-almost prime.
%t A114436 Select[Range[250],PrimeOmega[(#(#+1))/2]==5&] (* _Harvey P. Dale_, Sep 14 2012 *)
%t A114436 Flatten[Position[Accumulate[Range[700]], _?(PrimeOmega[#]== 5 &)]] (* _Vincenzo Librandi_, Apr 09 2014 *)
%Y A114436 Cf. A000217, A001222, A014614, A069904.
%K A114436 easy,nonn
%O A114436 1,1
%A A114436 _Jonathan Vos Post_, Feb 13 2006
%E A114436 Corrected and extended by _Harvey P. Dale_, Apr 02 2011