cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A068443 Triangular numbers which are the product of two primes.

Original entry on oeis.org

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
Offset: 1

Views

Author

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

Keywords

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

Examples

			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.

Links

Crossrefs

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.
Cf. A075875.

Programs

  • 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 &]
Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)
  • 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

Formula

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009
a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Extensions

Edited by Robert G. Wilson v, Jul 08 2002
Definition corrected by Zak Seidov, Mar 09 2008

A095146 Even binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

Original entry on oeis.org

6, 10, 20, 28, 36, 56, 66, 70, 78, 84, 120, 126, 136, 190, 210, 220, 252, 276, 286, 300, 330, 364, 378, 406, 462, 496, 528, 560, 630, 666, 680, 780, 792, 816, 820, 924, 946, 990, 1128, 1140, 1176, 1326, 1330, 1378, 1540, 1596, 1716, 1770, 1820, 1830, 2002
Offset: 1

Views

Author

Robert G. Wilson v, May 29 2004

Keywords

Crossrefs

Cf. A007318, A006987, A047999, A095147.

Programs

  • Mathematica
    Take[ Select[ Union[ Flatten[ Table[ Binomial[n, k], {n, 2, 61}, {k, 2, n - 2}]]], EvenQ[ # ] &], 51]

A124000 Semiprimes in A006987(n), or semiprime binomial coefficients: C(n,k), 2 <= k <= n-2.

Original entry on oeis.org

6, 10, 15, 21, 35, 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

Views

Author

Alexander Adamchuk, Oct 31 2006

Keywords

Comments

Conjecture: all a(n) except a(1) = 6 and a(2) = 10 are odd. Conjecture: all a(n) except a(5) = 35 are triangular numbers of the form p*(2p +/- 1) that belong to A068443(n) = {6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, ...} Triangular numbers with two distinct prime factors.
Besides 35 & 371953, all members were found by C(n, 2). - Robert G. Wilson v, Sep 16 2016
Of C(n,k), n: 4, 5, 6, 7, 11, 14, 23, 38, 47, 59, 62, 74, 83, 107, 158, 167, 179, 194, ..., . - Robert G. Wilson v, Sep 16 2016

Examples

			C(5,2) = 5!/(3!*2!) = 120/(6*2) = 10 is a semiprime (A001358), so 10 is in the sequence. - _Michael B. Porter_, Sep 17 2016

Links

Crossrefs

Cf. A006987, A095147, A068443, A000217, A005382, A005384, A001358.

Programs

  • Mathematica
    s = {}; Do[b = Binomial[n, k]; If[PrimeOmega@ b == 2, AppendTo[s, b]; Print@ b], {n, 3, 10000}, {k, 2, n/2}]; s (* Robert G. Wilson v, Nov 03 2006; updated Sep 16 2016 *)

Formula

Intersection of A001358 and A006987. - Michael B. Porter, Sep 17 2016

Extensions

More terms from Robert G. Wilson v, Nov 03 2006
Showing 1-3 of 3 results.

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