A102461 -id:A102461 - cp's OEIS frontend

A027568 Numbers that are both triangular and tetrahedral.

0, 1, 10, 120, 1540, 7140

Offset: 1

From Anthony C Robin, Oct 27 2022: (Start)
For numbers to be triangular and tetrahedral, we look for solutions r*(r+1)*(r+2)/6 = t*(t+1)/2 = a(n). The corresponding r and t are r = A224421(n-1) and t = A102349(n).
Writing m=r+1 and s=2t+1, this problem is equivalent to solving the Diophantine equation 3 + 4*(m^3 - m) = 3*s^2. The integer solutions for this equation are m = 0, 1, 2, 4, 9, 21, 35 and the corresponding values of s are 1, 1, 3, 9, 31, 111, 239. (End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Ellipses (Paris), 2008 (entry 10, page 3; entry 120, page 41).
L. J. Mordell, Diophantine Equations, Ac. Press, page 258.
P. Odifreddi, Il museo dei numeri, Rizzoli, 2014, page 224.
J. Roberts, The Lure of the Integers, page 53.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 21.

E. T. Avanesov, Solution of a problem on figurate numbers (in Russian), Acta Arith. 12 1966/1967 pages 409-420.
Patrick De Geest, Palindromic Tetrahedrals
M. Gardner, Letter to N. J. A. Sloane, circa Aug 11 1980, concerning A001110, A027568, A039596, etc.
J. Roberts, The Lure of the Integers, pp. 53. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Tetrahedral Number

Intersection of A000217 and A000292.

Cf. A102349, A102461, A102772, A224421.

{seq(binomial(i,3),i=0..100000) } intersect {seq(binomial(k,2), k= 0..100000)}; # Zerinvary Lajos, Apr 26 2008

With[{trno=Accumulate[Range[0,1000]]},Intersection[trno,Accumulate[ trno]]] (* Harvey P. Dale, May 25 2014 *)

for(n=0,1e3,if(ispolygonal(t=n*(n+1)*(n+2)/6,3),print1(t", "))) \\ Charles R Greathouse IV, Apr 07 2013

A303615 Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.

Original entry on oeis.org

-29, -25, -20, -14, -5, 5, 14, 16, 20, 25, 49, 70, 79, 130, 250, 305, 400, 695, 1555, 1645, 18895

Offset: 1

Tomohiro Yamada, May 29 2018

This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points in this curve.

This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).

			a(6) = 5: 5^3 - 525 * 5 + 10156 = 7656 = 88 * 87.

Roelof J. Stroeker and Benjamin M. M. de Weger, Elliptic binomial diophantine equations, Math. Comp. 68 (1999), 1257-1281.

Cf. A029728 (the complete list of solutions x to y^2=x^3+17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).

A301515 Complete list of integers x > 1 such that x^2 - x = y^q - y, where q is an odd prime and y is a prime power.

Original entry on oeis.org

3, 6, 16, 91, 280

Offset: 1

Tomohiro Yamada, Dec 15 2018

The corresponding values of (y, q) are (2, 3), (2, 5), (3, 5), (2, 13) and (5, 7). Mignotte and Pethő proved that the list is complete.
If we relax the condition that y should be a prime power, the equation x^2 - x = y^q - y has additionally two solutions (x, y, q) = (15, 6, 3) and (4930, 30, 5) (Fielder and Alford, 1998).
The result of Mordell (1963) implies that x^2 - x = y^3 - y has only three positive integral solutions (x, y) = (1, 1), (3, 2) and (15, 6).
Bugeaud, Mignotte, Siksek, Stoll and Tengely proved that (x, y) = (1, 1), (6, 2), (16, 3), (4930, 30) are the only positive integral solutions to x^2 - x = y^5 - y.
The equation x^p - x = y^q - y, with p, q odd primes and x,y > 1 has a solution 13^3 - 13 = 3^7 - 3 but no other solution is known.

			a(3) = 16: 16^2 - 16 = 240 = 3^5 - 3.

Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), 859-885.
Daniel C. Fielder and Cecil O. Alford, Observations from computer experiments on an integer equation, Applications of Fibonacci numbers, edited by G. E. Bergum, A. N. Philippou and A. F. Horadam, vol. 7, pp. 93-103.
M. Mignotte and A. Pethő, On the diophantine equation x^p - x = y^q - y, Publ. Mat. 43 (1999), 207-216.
L. J. Mordell, On the integer solutions of y(y+1)=x(x+1)(x+2), Pacific J. Math. 13 (1963), 1347-1351.

Cf. A102461 (the complete list of solutions x to (x^2 - x)/2 = (y^3 - y)/6).

r[x_, q_] := {x, y, q} /. {ToRules @ Reduce[y >= 2 && x^2 - x == y^q - y, y, Integers]};
r[x_] := Select[Table[r[x, q], {q, NextPrime[Log[2, x^2 - x + 2]]}] /. {{a_, b_, c_}} -> {a, b, c}, PrimeNu[#[[2]]]==1 && #[[3]] > 2&];
T = Table[r[x], {x, 2, 300}];
For[k = 1, k <= Length[T], k++, t = T[[k]]; If[t != {}, Print["x = ", t[[1, 1]], ", y = ", t[[1, 2]], ", q = ", t[[1, 3]]]]] (* Jean-François Alcover, Dec 17 2018 *)

A317053 Complete list of solutions to y^2 + y = x^3 - 525x + 10156 with y > 0; sequence gives positive y values.

Original entry on oeis.org

31, 74, 76, 87, 112, 121, 319, 562, 679, 1462, 3937, 5312, 7987, 18312, 61312, 66712, 2597287

Offset: 1

Tomohiro Yamada, Jul 20 2018

This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points with y > 0 in this curve.
Each positive y corresponds to a negative solution -y - 1, so that the sequence gives all y values of solutions.
Some y values corresponds to three solutions. For y = 87, we have x = -25, 5 or 20. For y = 112, we have x = -20, -5 or 25. Any other value of y corresponds to a unique solution.
This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).

			a(1) = 31: (-29)^3 - 525 * (-29) + 10156 = 996 = 31 * 32.

Roelof J. Stroeker and Benjamin M. M. de Weger, Elliptic binomial diophantine equations, Math. Comp. 68 (1999), 1257-1281.

Cf. A303615 (x values)

Cf. A029728 (the complete list of solutions x to y^2 = x^3 + 17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).

EllipticCurve([0, 0, 1, -525, 10156]).integral_points()

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A027568 Numbers that are both triangular and tetrahedral.

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A303615 Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.

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A301515 Complete list of integers x > 1 such that x^2 - x = y^q - y, where q is an odd prime and y is a prime power.

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A317053 Complete list of solutions to y^2 + y = x^3 - 525x + 10156 with y > 0; sequence gives positive y values.

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