A068443 - OEIS

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, 258121, 269011

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OFFSET

1,1

COMMENTS

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

All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - 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, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]

(* Alternative: *)

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

(Python)

from itertools import islice

from sympy import isprime, nextprime

def A068443_gen(): # generator of terms

p = 3

while True:

if isprime(p>>1):

yield (p-1)*p>>1

if isprime(p+1>>1):

yield p*(p+1)>>1

p = nextprime(p)

A068443_list = list(islice(A068443_gen(), 30)) # Chai Wah Wu, May 28 2026