A108815 Indices of triangular numbers which are products of 3 primes.

7, 9, 11, 12, 14, 17, 18, 19, 21, 25, 28, 29, 30, 33, 34, 38, 41, 42, 43, 52, 57, 66, 67, 70, 78, 85, 86, 93, 94, 97, 101, 102, 109, 113, 118, 121, 122, 130, 133, 137, 138, 141, 142, 145, 148, 158, 163, 172, 173, 177, 181, 190, 201, 202, 205, 211, 213, 214, 217, 218

Offset

1,1

Comments

Indices of 3-almost prime triangular numbers.

Links

T. D. Noe, 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)) = 3}. {a(n)} = {k such that k*(k+1)/2 has exactly 3 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014612}.

n such that n*(n+1)/2 is an element of A014612. n such that A000217(n) is an element of A014612. n such that C(n+1, 2) is an element of A014612.

{ m : A069904(m) = 3 }. - Alois P. Heinz, Aug 05 2019

Example

a(1) = 7 because T(7) = TriangularNumber(7) = 7*(7+1)/2 = 28 = 2^2 * 7 is a 3-almost prime.
a(2) = 9 because T(9) = 9*(9+1)/2 = 45 = 3^2 * 5 is a 3-almost prime.
a(3) = 11 because T(11) = 11*(11+1)/2 = 66 = 2 * 3 * 11.
a(31) = 101 because T(101) = 101*(101+1)/2 = 5151 = 3 * 17 * 101.
a(49) = 173 because T(173) = 173*(173+1)/2 = 15051 = 3 * 29 * 173.

Mathematica

Select[Range[225], Plus @@ Last /@ FactorInteger[ #*(# + 1)/2] == 3 &] (* Ray Chandler, Jul 16 2005 *)

Prog

(PARI) issemi(n)=bigomega(n)==2
is(n)=if(isprime(n/gcd(n, 2)), issemi((n+1)/gcd(n+1, 2)), isprime((n+1)/gcd(n+1, 2)) && issemi(n/gcd(n, 2))) \\ Charles R Greathouse IV, Feb 05 2017

Crossrefs

Cf. A000217, A001222, A014612, A069904.

Sequence in context: A283056 A162308 A191883 * A262536 A161992 A334102

Adjacent sequences: A108812 A108813 A108814 * A108816 A108817 A108818

Keyword

easy,nonn

Author

Jonathan Vos Post, Jul 10 2005

Extensions

Extended by Ray Chandler, Jul 16 2005

Edited by N. J. A. Sloane, May 07 2007

Status

approved