A068443 Triangular numbers which are the product of two primes.

6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801

Offset

1,1

Comments

These triangular numbers are equal to p * (2p +- 1).

All a(n) belong to A006987(n) = {6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, ...} Binomial coefficients: C(n,k), 2 <= k <= n-2. For n>2 all a(n) are odd and belong to A095147(n) = {15, 21, 35, 45, 55, 91, 105, 153, 165, 171, 231, 253, ...} Odd binomial coefficients: C(n,k), 2 <= k <= n-2. - Alexander Adamchuk, Oct 31 2006

A156592 is a subsequence. - Reinhard Zumkeller, Feb 10 2009

Triangular numbers with exactly 4 divisors. - Jon E. Schoenfield, Sep 05 2018

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Formula

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Example

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 *)
a(11) = 1891 and 1891 = 31 * 61.

Maple

q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024

Mathematica

Select[ Table[ n(n + 1)/2, {n, 700}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &].
Select[Accumulate[Range[1000]], PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)

Prog

(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

Crossrefs

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.

Cf. A075875.

Sequence in context: A124000 A229321 A229323 * A113940 A315280 A315281

Adjacent sequences: A068440 A068441 A068442 * A068444 A068445 A068446

Keyword

easy,nonn

Author

Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002

Extensions

Edited by Robert G. Wilson v, Jul 08 2002

Definition corrected by Zak Seidov, Mar 09 2008

Status

approved