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 1-5 of 5 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

A128896 Triangular numbers that are products of three distinct primes.

Original entry on oeis.org

66, 78, 105, 190, 231, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051, 15753
Offset: 1

Views

Author

Zak Seidov, Apr 20 2007

Keywords

Examples

			a(1)=T(11)=66=2*3*11, a(2)=T(12)=78=2*3*13, a(3)=T(14)=105=3*5*7, a(4)=T(19)=190=2*5*19, a(5)=T(21)=231=3*7*11, a(6)=T(28)=406=2*7*29.
T(15) = 120 = 2^3*3*5. The triangular 120 has three prime factors but is not a product of these factors. Thus, 120 is not in this sequence.

Links

Crossrefs

Cf. A000217, A068443, A069903, A076551, A127637.

Programs

  • Mathematica
    Select[Table[n(n+1)/2,{n,1,210}],Transpose[FactorInteger[ # ]][[2]]=={1,1,1}&]
Select[Accumulate[Range[200]],PrimeNu[#]==PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 23 2017 *)

Formula

a(n) = T(k) = k*(k+1)/2 = p*q*r for some k,p,q,r, where T(k) is triangular number and p, q, r are distinct primes.
Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - R. J. Mathar, Apr 22 2007

Extensions

Name clarified by Tanya Khovanova, Sep 06 2022

A076578 Triangular numbers which are 4-almost primes.

Original entry on oeis.org

36, 136, 210, 276, 351, 666, 820, 1035, 1225, 1275, 1326, 1431, 1770, 1830, 1953, 2145, 2346, 2415, 2775, 2926, 3003, 3486, 3916, 4005, 4186, 4278, 5050, 5356, 5565, 6105, 6555, 6670, 6903, 7626, 8001, 8385, 8646, 9316, 9730, 10731, 11175, 11325, 11476, 11935
Offset: 1

Views

Author

Shyam Sunder Gupta, Oct 19 2002

Keywords

Examples

			36 is a term because it is a triangular number and 36 = 2*2*3*3, i.e., is a product of 4 prime factors so is a 4-almost prime.

Links

Crossrefs

Intersection of A000217 and A014613.
Cf. A068443, A075875.

Programs

  • Maple
    q:= n-> is(numtheory[bigomega](n)=4):
select(q, [i*(i+1)/2$i=0..200])[];  # Alois P. Heinz, Mar 27 2024
  • Mathematica
    q[n_] := PrimeOmega[n] == 4;
Select[Table[i*(i+1)/2, {i, 0, 200}], q] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

A101744 Triangular numbers which are 10-almost primes.

Original entry on oeis.org

32640, 73920, 130816, 165600, 204480, 265356, 294528, 401856, 592416, 839160, 947376, 990528, 1279200, 1445850, 1492128, 1606528, 1842240, 1844160, 2031120, 2049300, 2821500, 2956096, 3571128, 3963520, 4148640, 4250070, 4335040
Offset: 1

Views

Author

Jonathan Vos Post, Dec 14 2004

Keywords

Comments

A101745 contains the indices of this sequence, i.e. T(n) for what values of n are these 10-almost primes.

Examples

			a(1) = 32640 because that is the smallest triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

References

  • Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.
  • Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.
  • Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

Links

Crossrefs

Cf. A000217, A046314, A068443, A075875, A076578-A076583, A075088, A101745.

Programs

  • Mathematica
    BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Select[Table[n*(n+1)/2, {n, 2, 5000}], BigOmega[ # ]==10&] (* Ray Chandler, Dec 14 2004 *)
  • PARI
    list(lim)=my(v=List(),cur,last=3,n=256,t); while((t=n*(n-1)/2) <= lim, cur=bigomega(n); if(cur+old==11, listput(v,t)); old=cur; n++); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

Formula

a(n) is in the intersection of {A000217} and {A046314}. Integers of the form k*(k+1)/2 which have exactly 10 prime factors.

Extensions

More terms from Ray Chandler, Dec 14 2004

A255200 Numbers n such that triangular numbers T(n), T(n+1) and T(n+2) are 3-almost primes.

Original entry on oeis.org

17, 28, 41, 281, 2081, 2801, 4721, 5441, 6196, 12161, 12916, 13996, 20476, 24916, 32321, 32441, 34156, 34961, 40036, 56596, 70996, 73361, 94396
Offset: 1

Views

Author

Zak Seidov, Feb 16 2015

Keywords

Comments

Numbers n such that n, n+1 and n+2 are terms in A108815.

Examples

			T(17)=A000217(17)=153=3*3*17, T(18)=A000217(18)=171=3*3*19, T(19)=A000217(19)=171=2*5*19.

Links

Crossrefs

Cf. A000217, A075875, A108815.

Programs

  • PARI
    issemi(n)=bigomega(n)==2
is(n)=if(n%2, if(n%4==1, isprime((n+3)/4) && isprime(n+2) && isprime(n) && issemi((n+1)/2), isprime((n+1)/4) && isprime(n) && isprime(n+2) && issemi((n+3)/2)), if(n%4==2, isprime((n+2)/4) && isprime(n+1) && isprime(n+3) && issemi(n/2), isprime(n/4) && isprime(n+1) && isprime(n+3) && issemi(n/2+1))) \\ Charles R Greathouse IV, Feb 05 2017
Showing 1-5 of 5 results.

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