A003556 -id:A003556

Search: a003556 -id:a003556

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A216268

Tetrahedral numbers of the form k^2 - 1.

+10
2

0, 35, 120, 2024, 2600, 43680, 435730689800

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..7.

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 *)

PROG

(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=' ')

(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 */

CROSSREFS

Cf. A000292, A216269, A175492.

Cf. A003556 (both square and tetrahedral).

KEYWORD

nonn,more,fini

AUTHOR

Alex Ratushnyak, Mar 15 2013

STATUS

approved

A216269

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..7.

Wikipedia, Siegel's theorem on integral points

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 *)

PROG

(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)

CROSSREFS

Cf. A000292, A216268.

Cf. A003556 (both square and tetrahedral).

KEYWORD

nonn,fini

AUTHOR

Alex Ratushnyak, Mar 15 2013

STATUS

approved

A361671

Squarefree part of the n-th tetrahedral number.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..58.

FORMULA

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

PROG

(Python)

from sympy.ntheory.factor_ import core

def A361671(n): return core(n*(n*(n + 3) + 2)//6) # Chai Wah Wu, Mar 20 2023

(PARI) a(n) = core(n*(n+1)*(n+2)/6); \\ Michel Marcus, Mar 22 2023

CROSSREFS

Cf. A007913, A000292, A361670 (of triangular), A083481 (of oblong).

Cf. A003556 (squarefree part is 1).

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Mar 20 2023

STATUS

approved

A307491

Numbers that are both centered triangular and tetrahedral.

+10
1

1, 4, 10, 4960, 428536

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..5.

Eric Weisstein's World of Mathematics, Centered Triangular Number

Eric Weisstein's World of Mathematics, Tetrahedral Number

CROSSREFS

Intersection of A000292 and A005448.

Cf. A003556, A027568, A128862, A131750, A307492.

KEYWORD

nonn,more

AUTHOR

Ilya Gutkovskiy, Apr 10 2019

STATUS

approved

A216267

Numbers that are both tetrahedral and pronic.

+10
0

0, 20, 56, 7140, 1414910

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Intersection of A000292 and A002378.

LINKS

Table of n, a(n) for n=1..5.

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 *)

PROG

(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)

CROSSREFS

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

KEYWORD

nonn,more

AUTHOR

Alex Ratushnyak, Mar 15 2013

STATUS

approved

A353064

Numbers simultaneously square and heptagonal pyramidal.

+10
0

0, 1, 196, 99225

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Is this sequence finite?

No other terms < 10^32. - Michael S. Branicky, Jul 12 2022

LINKS

Table of n, a(n) for n=1..4.

ProofWiki, Heptagonal Pyramidal Numbers which are Square

EXAMPLE

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.

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 *)

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).

KEYWORD

nonn,more

AUTHOR

Kelvin Voskuijl, Apr 21 2022

STATUS

approved

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