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.

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A180590 Numbers k such that k! is the sum of two triangular numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16, 17, 21, 24, 27, 28, 29, 32, 33, 34, 42, 49, 54, 59, 66, 68, 72, 79, 80, 81, 85, 86, 95, 96, 99, 102, 109, 118
Offset: 1

Views

Author

Robert G. Wilson v, Sep 10 2010

Keywords

Comments

Numbers k such that there are nonnegative numbers x and y such that x*(x+1)/2 + y*(y+1)/2 = k!. Equivalently, (2x+1)^2 + (2y+1)^2 = 8k! + 2. A necessary and sufficient condition for this is that all the prime factors of 4k!+1 that are congruent to 3 (mod 4) occur to even powers (cf. A001481).
Based on an email from R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010.
See A152089 for further links.

Examples

			0! = 1! = T(0) + T(1);
2! = T(1) + T(1);
3! = T(0) + T(3) = T(2) + T(2);
4! = T(2) + T(6);
5! = T(0) + T(15) = T(5) + T(14);
7! = T(45) + T(89);
8! = T(89) + T(269);
9! = T(210) + T(825);
10! = T(665) + T(2610) = T(1770) + T(2030);
13! = T(71504) + T(85680);
15! = T(213384) + T(1603064) = T(299894) + T(1589154);
16! = T(3631929) + T(5353005);
17! = T(12851994) + T(23370945) = T(17925060) + T(19750115);
etc.

Links

Crossrefs

Complement of A152089. A171099 gives the number of solutions.
Cf. A000142, A000217, A051533, A076680.

Programs

  • Mathematica
    triQ[n_] := IntegerQ@ Sqrt[8 n + 1]; fQ[n_] := Block[{k = 1, lmt = Floor@Sqrt[2*n! ], nf = n!}, While[k < lmt && ! triQ[nf - k (k + 1)/2], k++ ]; r = (Sqrt[8*(nf - k (k + 1)/2) + 1] - 1)/2; Print[{k, r, n}]; If[IntegerQ@r, True, False]]; k = 1; lst = {}; While[k < 69, If[ fQ@ k, AppendTo[lst, k]]; k++ ]; lst
  • Python
    from math import factorial
from itertools import count, islice
from sympy import factorint
def A180590_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(4*factorial(n)+1).items()),count(0))
A180590_list = list(islice(A180590_gen(),15)) # Chai Wah Wu, Jun 27 2022

Extensions

Edited by N. J. A. Sloane, Sep 24 2010
69 eliminated (see A152089) by N. J. A. Sloane, Sep 24 2010
Extended by Georgi Guninski and D. S. McNeil, Sep 24 2010
a(35)-a(38) from Georgi Guninski, Oct 12 2010
a(39)-a(40) from Tyler Busby, Apr 24 2025

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x*(x+1)/2 + y*(y+1)/2 = n!.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 8, 1, 2, 0, 0, 4, 4, 16, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 1, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 2, 4, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

N. J. A. Sloane, Sep 24 2010, based on a posting by R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010

Keywords

Examples

			Initial solutions: (x,y,n) = (0,1,0), (0,1,1), (1,1,2), (0,3,3), (2,2,3), (2,6,4), (0,15,5), (5,14,5), (45,89,7), (89,269,8), (210,825,9), (760,2610,10), (1770,2030,10), none for n = 11 or 12, one for n = 13 (71504,85680,13) (found by _Ed Pegg Jr_), etc.

Crossrefs

Cf. A000161, A152089 (n for which no solutions exist), A180590 (n for which solutions exist).

Formula

a(n) = A000161(8*n! + 2). - Max Alekseyev, Dec 12 2011

Extensions

Corrected and extended (with data from Georgi Guninski, at the suggestion of N. J. A. Sloane) by D. S. McNeil, Sep 26 2010
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