A114435 Indices of 4-almost prime triangular numbers.

8, 16, 20, 23, 26, 36, 40, 45, 49, 50, 51, 53, 59, 60, 62, 65, 68, 69, 74, 76, 77, 83, 88, 89, 91, 92, 100, 103, 105, 110, 114, 115, 117, 123, 126, 129, 131, 136, 139, 146, 149, 150, 151, 154, 156, 165, 169, 182, 185, 186, 187, 194, 196, 197, 198, 206, 210

Offset: 1

Author

Jonathan Vos Post, Feb 13 2006

Keywords

• easy
• nonn

Examples

			a(1) = 8 because T(8) = TriangularNumber(8) = 8*(8+1)/2 = 36 = 2^2 * 3^2 is a 4-almost prime.
a(2) = 16 because T(16) = 16*(16+1)/2 = 136 = 2^3 * 17 is a 4-almost prime.
a(3) = 20 because T(20) = 20*(20+1)/2 = 210 = 2 * 3 * 5 * 7 (210 = primorial 4#).
a(4) = 23 because T(23) = 23*(23+1)/2 = 276 = 2^2 * 3 * 23.
a(5) = 26 because T(26) = 26*(26+1)/2 = 351 = 3^3 * 13.
a(6) = 36 because T(36) = 36*(36+1)/2 = 666 = 2 * 3^2 * 37.
a(27) = 100 because T(100) = 100*(100+1)/2 = 5050 = 2 * 5^2 * 101.
a(57) = 210 because T(210) = 210*(210+1)/2 = 22155 = 3 * 5 * 7 * 211 (again, 210 = primorial 4#).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Triangular Number.
• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

Cf. A000217, A001222, A014613, A069904.

Programs

Mathematica

Flatten[Position[Accumulate[Range[800]], ?(PrimeOmega[#]== 4 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

PARI

is(n)=my(t=bigomega(n/gcd(n,2))); if(t<3, bigomega((n+1)/gcd(n+1,2))+t==4, t==3 && isprime((n+1)/gcd(n+1,2))) \\ Charles R Greathouse IV, Jun 14 2017

Formula

{a(n)} = {k such that A001222(A000217(k)) = 4}. {a(n)} = {k such that k*(k+1)/2 has exactly 4 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014613}.

{ m : A069904(m) = 4 }. - Alois P. Heinz, Aug 05 2019