A114437 - OEIS

%I #32 Mar 01 2023 06:41:49

%S 32,48,96,99,104,111,119,120,125,152,161,168,176,188,189,195,200,208,

%T 223,231,239,240,252,260,264,275,299,300,303,304,315,336,342,343,344,

%U 352,359,363,374,377,391,392,395,400

%N Indices of 6-almost prime triangular numbers.

%H Vincenzo Librandi, <a href="/A114437/b114437.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F {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}.

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

%F { m :  A069904(m) = 6 }. - _Alois P. Heinz_, Aug 05 2019

%e a(1) = 48 because T(48) = TriangularNumber(48) = 48*(48+1)/2 = 1176 = 2^3 * 3 * 7^2 is a 6-almost prime.

%e a(2) = 96 because T(96) = 96*(96+1)/2 = 4656 = 2^4 * 3 * 97 is a 6-almost prime.

%e a(18) = 200 because T(200) = 200*(200+1)/2 = 20100 = 2^2 * 3 * 5^2 * 67 is a 6-almost prime.

%e a(29) = 300 because T(300) = 300*(300+1)/2 = 45150 = 2 * 3 * 5^2 * 7 * 43 is a 6-almost prime.

%e a(38) = 363 because T(363) = 363*(363+1)/2 = 45150 = 66066 = 2 * 3 * 7 * 11^2 * 13 is a 6-almost prime.

%t Select[Range[400],PrimeOmega[(#(#+1))/2]==6&] (* _Harvey P. Dale_, Mar 29 2012 *)

%t Flatten[Position[Accumulate[Range[800]], _?(PrimeOmega[#]== 6 &)]] (* _Vincenzo Librandi_, Apr 09 2014 *)

%o (PARI) isA114437(n)=bigomega(n*(n+1)/2)==6 /* _Michael B. Porter_, Mar 30 2012 */

%Y Cf. A000217, A001222, A046306, A069904.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 14 2006

%E Corrected by _Harvey P. Dale_, Mar 29 2012