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 A068443 Triangular numbers which are the product of two primes. 17
 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 (list; graph; refs; listen; history; text; internal format)
 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 A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 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 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. MATHEMATICA Select[ Table[ n(n + 1)/2, {n, 700}], Apply[Plus, Transpose[ FactorInteger[ # ]] []] == 2 &]. Select[Accumulate[Range], 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. 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

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Last modified May 17 14:19 EDT 2022. Contains 353746 sequences. (Running on oeis4.)