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 1-6 of 6 results.

A216268 Tetrahedral numbers of the form k^2 - 1.

Original entry on oeis.org

0, 35, 120, 2024, 2600, 43680, 435730689800
Offset: 1

Views

Author

Alex Ratushnyak, Mar 15 2013

Keywords

Comments

This sequence is finite by Siegel's theorem on integral points. The next term, if it exists, is greater than 10^35. - David Radcliffe, Jan 01 2024

Crossrefs

Cf. A000292, A216269, A175492.
Cf. A003556 (both square and tetrahedral).

Programs

  • Maple
    select(t -> issqr(t+1), [seq(i*(i+1)*(i+2)/6, i=0..10^6)]); # Robert Israel, Jan 02 2024
  • Mathematica
    t = {}; Do[tet = n (n + 1) (n + 2)/6; If[IntegerQ[Sqrt[tet + 1]], AppendTo[t, tet]], {n, 0, 100000}]; t (* T. D. Noe, Mar 18 2013 *)
  • PARI
    A000292(n) = n*(n+1)*(n+2)\6;
for(n=0,10^9, t=A000292(n); if (issquare(t+1), print1(t,", ") ) );
/* Joerg Arndt, Mar 16 2013 */
  • Python
    import math
for i in range(1<<33):
    t = i*(i+1)*(i+2)//6 + 1
    sr = math.isqrt(t)
    if sr*sr == t:
        print (t-1, sep=' ')

A216269 Numbers n such that n^2 - 1 is a tetrahedral number (A000292).

Original entry on oeis.org

1, 6, 11, 45, 51, 209, 660099
Offset: 1

Views

Author

Alex Ratushnyak, Mar 15 2013

Keywords

Comments

Corresponding tetrahedral numbers are in A216268.
The curve 6*(x^2-1)-y*(y+1)*(y+2)=0 is elliptic, and has finitely many integral points by Siegel's theorem. - Robert Israel, Apr 22 2021

Links

Crossrefs

Cf. A000292, A216268.
Cf. A003556 (both square and tetrahedral).

Programs

  • Mathematica
    t = {}; Do[tet = n (n + 1) (n + 2)/6; If[IntegerQ[s = Sqrt[tet + 1]], AppendTo[t, s]], {n, 0, 100000}]; t (* T. D. Noe, Mar 18 2013 *)
  • Python
    import math
for i in range(1<<30):
    t = i*(i+1)*(i+2)//6 + 1
    sr = int(math.sqrt(t))
    if sr*sr == t:
        print(sr)

A361671 Squarefree part of the n-th tetrahedral number.

Original entry on oeis.org

1, 1, 10, 5, 35, 14, 21, 30, 165, 55, 286, 91, 455, 35, 170, 51, 969, 285, 1330, 385, 1771, 506, 23, 26, 13, 91, 406, 1015, 4495, 310, 341, 374, 6545, 1785, 7770, 2109, 9139, 2470, 2665, 2870, 12341, 3311, 14190, 3795, 16215, 1081, 94, 1, 17, 221, 23426, 689, 2915, 770, 7315, 7714, 32509, 8555
Offset: 1

Views

Author

R. J. Mathar, Mar 20 2023

Keywords

Crossrefs

Cf. A007913, A000292, A361670 (of triangular), A083481 (of oblong).
Cf. A003556 (squarefree part is 1).

Programs

  • PARI
    a(n) = core(n*(n+1)*(n+2)/6); \\ Michel Marcus, Mar 22 2023
  • Python
    from sympy.ntheory.factor_ import core
def A361671(n): return core(n*(n*(n + 3) + 2)//6) # Chai Wah Wu, Mar 20 2023

Formula

a(n) = A007913(A000292(n)).

A307491 Numbers that are both centered triangular and tetrahedral.

Original entry on oeis.org

1, 4, 10, 4960, 428536
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

Comments

If it exists, a(6) > 10^29. - Bert Dobbelaere, Apr 12 2019

Links

Crossrefs

Intersection of A000292 and A005448.
Cf. A003556, A027568, A128862, A131750, A307492.

A216267 Numbers that are both tetrahedral and pronic.

Original entry on oeis.org

0, 20, 56, 7140, 1414910
Offset: 1

Views

Author

Alex Ratushnyak, Mar 15 2013

Keywords

Comments

Intersection of A000292 and A002378.
The equation y*(y+1) = x*(x+1)*(x+2)/6 leads to an elliptic curve, which has a finite number of solutions, all of which are already listed. - Max Alekseyev, Dec 28 2024

Crossrefs

Cf. A000292, A002378, A027568, A029549, A003556.

Programs

  • Mathematica
    t = {}; Do[tet = n (n + 1) (n + 2)/6; s = Floor[Sqrt[tet]]; If[s^2 + s == tet, AppendTo[t, tet]], {n, 0, 1000}]; t (* T. D. Noe, Mar 18 2013 *)
With[{nn=50000},Intersection[Binomial[Range[0,nn]+2,3],Table[n(n+1),{n,nn}]]] (* Harvey P. Dale, Apr 04 2016 *)
  • Python
    def rootPronic(a):
    sr = 1<<33
    while a < sr*(sr+1):
      sr>>=1
    b = sr>>1
    while b:
        s = sr+b
        if a >= s*(s+1):
          sr = s
        b>>=1
    return sr
for i in range(1<<20):
      a = i*(i+1)*(i+2)//6
      t = rootPronic(a)
      if a == t*(t+1):
        print(a)

Extensions

fini, full keywords added by Max Alekseyev, Dec 28 2024

A353064 Numbers simultaneously square and heptagonal pyramidal.

Original entry on oeis.org

0, 1, 196, 99225
Offset: 1

Views

Author

Kelvin Voskuijl, Apr 21 2022

Keywords

Comments

Is this sequence finite?
No other terms < 10^32. - Michael S. Branicky, Jul 12 2022

Examples

			196 is a term because 196 = 14^2 is a perfect square and 196 = 6*(6+1)*(5*6-2)/6 is the 6th heptagonal pyramidal number.

Links

Crossrefs

Intersection of A000290 and A002413.
Cf. A003556 (tetrahedral and square), 1 and 4900 are only squares that are square pyramidal, A277792 (pentagonal pyramidal and square).

Programs

  • Maple
    select(issqr, [seq(n*(n+1)*(5*n-2)/6, n=0..50)])[];  # Alois P. Heinz, Apr 21 2022
  • Mathematica
    Select[Table[n*(n + 1)*(5*n - 2)/6, {n, 0, 100}], IntegerQ @ Sqrt[#] &] (* Amiram Eldar, Apr 21 2022 *)
Showing 1-6 of 6 results.

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