

A068443


Triangular numbers which are the product of two primes.


19



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
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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 <= n2. 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 <= n2.  Alexander Adamchuk, Oct 31 2006


LINKS



FORMULA



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.


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*p1), listput(v, 2*p^2p)); 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, A157344A157347, A157352A157357, A164977.


KEYWORD

easy,nonn


AUTHOR

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


EXTENSIONS



STATUS

approved



