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A114435 Indices of 4-almost prime triangular numbers. 5

%I

%S 8,16,20,23,26,36,40,45,49,50,51,53,59,60,62,65,68,69,74,76,77,83,88,

%T 89,91,92,100,103,105,110,114,115,117,123,126,129,131,136,139,146,149,

%U 150,151,154,156,165,169,182,185,186,187,194,196,197,198,206,210

%N Indices of 4-almost prime triangular numbers.

%H Vincenzo Librandi, <a href="/A114435/b114435.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F {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}.

%F { m : A069904(m) = 4 }. - _Alois P. Heinz_, Aug 05 2019

%e a(1) = 8 because T(8) = TriangularNumber(8) = 8*(8+1)/2 = 36 = 2^2 * 3^2 is a 4-almost prime.

%e a(2) = 16 because T(16) = 16*(16+1)/2 = 136 = 2^3 * 17 is a 4-almost prime.

%e a(3) = 20 because T(20) = 20*(20+1)/2 = 210 = 2 * 3 * 5 * 7 (210 = primorial 4#).

%e a(4) = 23 because T(23) = 23*(23+1)/2 = 276 = 2^2 * 3 * 23.

%e a(5) = 26 because T(26) = 26*(26+1)/2 = 351 = 3^3 * 13.

%e a(6) = 36 because T(36) = 36*(36+1)/2 = 666 = 2 * 3^2 * 37.

%e a(27) = 100 because T(100) = 100*(100+1)/2 = 5050 = 2 * 5^2 * 101.

%e a(57) = 210 because T(210) = 210*(210+1)/2 = 22155 = 3 * 5 * 7 * 211 (again, 210 = primorial 4#).

%t Flatten[Position[Accumulate[Range[800]], _?(PrimeOmega[#]== 4 &)]] (* _Vincenzo Librandi_, Apr 09 2014 *)

%o (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

%Y Cf. A000217, A001222, A014613, A069904.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 13 2006

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)