A114437 Indices of 6-almost prime triangular numbers.

32, 48, 96, 99, 104, 111, 119, 120, 125, 152, 161, 168, 176, 188, 189, 195, 200, 208, 223, 231, 239, 240, 252, 260, 264, 275, 299, 300, 303, 304, 315, 336, 342, 343, 344, 352, 359, 363, 374, 377, 391, 392, 395, 400

Offset: 1

Author

Jonathan Vos Post, Feb 14 2006

Keywords

• easy
• nonn

Examples

			a(1) = 48 because T(48) = TriangularNumber(48) = 48*(48+1)/2 = 1176 = 2^3 * 3 * 7^2 is a 6-almost prime.
a(2) = 96 because T(96) = 96*(96+1)/2 = 4656 = 2^4 * 3 * 97 is a 6-almost prime.
a(18) = 200 because T(200) = 200*(200+1)/2 = 20100 = 2^2 * 3 * 5^2 * 67 is a 6-almost prime.
a(29) = 300 because T(300) = 300*(300+1)/2 = 45150 = 2 * 3 * 5^2 * 7 * 43 is a 6-almost prime.
a(38) = 363 because T(363) = 363*(363+1)/2 = 45150 = 66066 = 2 * 3 * 7 * 11^2 * 13 is a 6-almost prime.
		

Links

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

Crossrefs

Cf. A000217, A001222, A046306, A069904.

Programs

Mathematica

Select[Range[400],PrimeOmega[(#(#+1))/2]==6&] (* Harvey P. Dale, Mar 29 2012 *)
Flatten[Position[Accumulate[Range[800]], ?(PrimeOmega[#]== 6 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

PARI

isA114437(n)=bigomega(n*(n+1)/2)==6 /* Michael B. Porter, Mar 30 2012 */

Formula

{a(n)} = {k such that A001222(A000217(k)) = 6}. {a(n)} = {k such that k*(k+1)/2 has exactly 6 prime factors, with multiplicity}.

{a(n)} = {k such that A000217(k) is an element of A046306}.

{ m : A069904(m) = 6 }. - Alois P. Heinz, Aug 05 2019

Extensions

Corrected by Harvey P. Dale, Mar 29 2012