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

1,1

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.

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

Example

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#).

Mathematica

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

Prog

(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

Crossrefs

Cf. A000217, A001222, A014613, A069904.

Sequence in context: A232570 A029522 A033309 * A327092 A112679 A212021

Adjacent sequences: A114432 A114433 A114434 * A114436 A114437 A114438

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 13 2006

Status

approved