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Indices of 5-almost prime triangular numbers.
5

%I #17 Aug 05 2019 18:13:48

%S 15,24,27,31,35,39,44,47,54,55,56,71,72,75,79,81,84,87,90,98,107,108,

%T 112,116,124,132,134,140,147,153,155,162,164,167,170,171,174,179,180,

%U 183,184,199,203,204,209,219,220,225,230,234,244,245,247,248,249

%N Indices of 5-almost prime triangular numbers.

%H Vincenzo Librandi, <a href="/A114436/b114436.txt">Table of n, a(n) for n = 1..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>

%F {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 { m : A069904(m) = 5 }. - _Alois P. Heinz_, Aug 05 2019

%e a(1) = 15 because T(15) = TriangularNumber(15) = 15*(15+1)/2 = 120 = 2^3 * 3 * 5 is a 5-almost prime.

%e a(2) = 24 because T(24) = 24*(24+1)/2 = 300 = 2^2 * 3 * 5^2 is a 5-almost prime.

%e a(3) = 27 because T(27) = 27*(27+1)/2 = 378 = 2 * 3^3 * 7 is a 5-almost prime.

%e a(4) = 31 because T(27) = 31*(31+1)/2 = 496 = 2^4 * 31 is a 5-almost prime.

%e a(17) = 84 because T(27) = 84*(84+1)/2 = 3570 = 2 * 3 * 5 * 7 * 17 is a 5-almost prime.

%t Select[Range[250],PrimeOmega[(#(#+1))/2]==5&] (* _Harvey P. Dale_, Sep 14 2012 *)

%t Flatten[Position[Accumulate[Range[700]], _?(PrimeOmega[#]== 5 &)]] (* _Vincenzo Librandi_, Apr 09 2014 *)

%Y Cf. A000217, A001222, A014614, A069904.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 13 2006

%E Corrected and extended by _Harvey P. Dale_, Apr 02 2011