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A068443
Triangular numbers which are the product of two primes.
21
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
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