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A114437 Indices of 6-almost prime triangular numbers.

%I A114437 #33 Feb 16 2025 08:32:59
%S A114437 32,48,96,99,104,111,119,120,125,152,161,168,176,188,189,195,200,208,
%T A114437 223,231,239,240,252,260,264,275,299,300,303,304,315,336,342,343,344,
%U A114437 352,359,363,374,377,391,392,395,400
%N A114437 Indices of 6-almost prime triangular numbers.
%H A114437 Vincenzo Librandi, <a href="/A114437/b114437.txt">Table of n, a(n) for n = 1..1000</a>
%H A114437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.
%F A114437 {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 A114437 {a(n)} = {k such that A000217(k) is an element of A046306}.
%F A114437 { m :  A069904(m) = 6 }. - _Alois P. Heinz_, Aug 05 2019
%e A114437 a(1) = 48 because T(48) = TriangularNumber(48) = 48*(48+1)/2 = 1176 = 2^3 * 3 * 7^2 is a 6-almost prime.
%e A114437 a(2) = 96 because T(96) = 96*(96+1)/2 = 4656 = 2^4 * 3 * 97 is a 6-almost prime.
%e A114437 a(18) = 200 because T(200) = 200*(200+1)/2 = 20100 = 2^2 * 3 * 5^2 * 67 is a 6-almost prime.
%e A114437 a(29) = 300 because T(300) = 300*(300+1)/2 = 45150 = 2 * 3 * 5^2 * 7 * 43 is a 6-almost prime.
%e A114437 a(38) = 363 because T(363) = 363*(363+1)/2 = 45150 = 66066 = 2 * 3 * 7 * 11^2 * 13 is a 6-almost prime.
%t A114437 Select[Range[400],PrimeOmega[(#(#+1))/2]==6&] (* _Harvey P. Dale_, Mar 29 2012 *)
%t A114437 Flatten[Position[Accumulate[Range[800]], _?(PrimeOmega[#]== 6 &)]] (* _Vincenzo Librandi_, Apr 09 2014 *)
%o A114437 (PARI) isA114437(n)=bigomega(n*(n+1)/2)==6 /* _Michael B. Porter_, Mar 30 2012 */
%Y A114437 Cf. A000217, A001222, A046306, A069904.
%K A114437 easy,nonn
%O A114437 1,1
%A A114437 _Jonathan Vos Post_, Feb 14 2006
%E A114437 Corrected by _Harvey P. Dale_, Mar 29 2012