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URL: https://oeis.org/A114435

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A114435
Indices of 4-almost prime triangular numbers.
5
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
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 13 2006
STATUS
approved