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A068443 Triangular numbers which are the product of two primes.

%I A068443 #55 Aug 06 2025 08:47:48
%S A068443 6,10,15,21,55,91,253,703,1081,1711,1891,2701,3403,5671,12403,13861,
%T A068443 15931,18721,25651,34453,38503,49141,60031,64261,73153,79003,88831,
%U A068443 104653,108811,114481,126253,146611,158203,171991,188191,218791,226801,258121,269011
%N A068443 Triangular numbers which are the product of two primes.
%C A068443 These triangular numbers are equal to p * (2p +- 1).
%C A068443 All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - _Alexander Adamchuk_, Oct 31 2006
%C A068443 A156592 is a subsequence. - _Reinhard Zumkeller_, Feb 10 2009
%C A068443 Triangular numbers with exactly 4 divisors. - _Jon E. Schoenfield_, Sep 05 2018
%H A068443 Jon E. Schoenfield, <a href="/A068443/b068443.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%F A068443 A010054(a(n))*A064911(a(n)) = 1. - _Reinhard Zumkeller_, Dec 03 2009
%F A068443 a(n) = A000217(A164977(n)). - _Zak Seidov_, Feb 16 2015
%e A068443 Triangular numbers begin 0, 1, 3, 6, 10, ...; 6=2*3, and 2 and 3 are two distinct primes; 10=2*5, and 2 and 5 are two distinct primes, etc. - _Vladimir Joseph Stephan Orlovsky_, Feb 27 2009
%e A068443 a(11) = 1891 and 1891 = 31 * 61.
%p A068443 q:= n-> is(numtheory[bigomega](n)=2):
%p A068443 select(q, [i*(i+1)/2$i=0..1000])[];  # _Alois P. Heinz_, Mar 27 2024
%t A068443 Select[ Table[ n(n + 1)/2, {n, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]
%t A068443 Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* _Harvey P. Dale_, Apr 03 2016 *)
%o A068443 (PARI) list(lim)=my(v=List());forprime(p=2,(sqrtint(lim\1*8+1)+1)\4, if(isprime(2*p-1),listput(v,2*p^2-p)); if(isprime(2*p+1), listput(v,2*p^2+p))); Vec(v) \\ _Charles R Greathouse IV_, Jun 13 2013
%Y A068443 Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.
%Y A068443 Cf. A075875.
%K A068443 easy,nonn
%O A068443 1,1
%A A068443 Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002
%E A068443 Edited by _Robert G. Wilson v_, Jul 08 2002
%E A068443 Definition corrected by _Zak Seidov_, Mar 09 2008