A114436 Indices of 5-almost prime triangular numbers.

15, 24, 27, 31, 35, 39, 44, 47, 54, 55, 56, 71, 72, 75, 79, 81, 84, 87, 90, 98, 107, 108, 112, 116, 124, 132, 134, 140, 147, 153, 155, 162, 164, 167, 170, 171, 174, 179, 180, 183, 184, 199, 203, 204, 209, 219, 220, 225, 230, 234, 244, 245, 247, 248, 249

Offset: 1

Author

Jonathan Vos Post, Feb 13 2006

Keywords

• easy
• nonn

Examples

			a(1) = 15 because T(15) = TriangularNumber(15) = 15*(15+1)/2 = 120 = 2^3 * 3 * 5 is a 5-almost prime.
a(2) = 24 because T(24) = 24*(24+1)/2 = 300 = 2^2 * 3 * 5^2 is a 5-almost prime.
a(3) = 27 because T(27) = 27*(27+1)/2 = 378 = 2 * 3^3 * 7 is a 5-almost prime.
a(4) = 31 because T(27) = 31*(31+1)/2 = 496 = 2^4 * 31 is a 5-almost prime.
a(17) = 84 because T(27) = 84*(84+1)/2 = 3570 = 2 * 3 * 5 * 7 * 17 is a 5-almost prime.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Eric Weisstein's World of Mathematics, Triangular Number.
• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

Cf. A000217, A001222, A014614, A069904.

Programs

Mathematica

Select[Range[250],PrimeOmega[(#(#+1))/2]==5&] (* Harvey P. Dale, Sep 14 2012 *)
Flatten[Position[Accumulate[Range[700]], ?(PrimeOmega[#]== 5 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

Formula

{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}.

{ m : A069904(m) = 5 }. - Alois P. Heinz, Aug 05 2019

Extensions

Corrected and extended by Harvey P. Dale, Apr 02 2011