A108815 - OEIS

%I #20 Feb 16 2025 08:32:58

%S 7,9,11,12,14,17,18,19,21,25,28,29,30,33,34,38,41,42,43,52,57,66,67,

%T 70,78,85,86,93,94,97,101,102,109,113,118,121,122,130,133,137,138,141,

%U 142,145,148,158,163,172,173,177,181,190,201,202,205,211,213,214,217,218

%N Indices of triangular numbers which are products of 3 primes.

%C Indices of 3-almost prime triangular numbers.

%H T. D. Noe, <a href="/A108815/b108815.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>

%F {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}.

%F 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.

%F { m :  A069904(m) = 3 }. - _Alois P. Heinz_, Aug 05 2019

%e a(1) = 7 because T(7) = TriangularNumber(7) = 7*(7+1)/2 = 28 = 2^2 * 7 is a 3-almost prime.

%e a(2) = 9 because T(9) = 9*(9+1)/2 = 45 = 3^2 * 5 is a 3-almost prime.

%e a(3) = 11 because T(11) = 11*(11+1)/2 = 66 = 2 * 3 * 11.

%e a(31) = 101 because T(101) = 101*(101+1)/2 = 5151 = 3 * 17 * 101.

%e a(49) = 173 because T(173) = 173*(173+1)/2 = 15051 = 3 * 29 * 173.

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

%o (PARI) issemi(n)=bigomega(n)==2

%o 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

%Y Cf. A000217, A001222, A014612, A069904.

%K easy,nonn,changed

%O 1,1

%A _Jonathan Vos Post_, Jul 10 2005

%E Extended by _Ray Chandler_, Jul 16 2005

%E Edited by _N. J. A. Sloane_, May 07 2007