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

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

1,3

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.

Links

Table of n, a(n) for n=1..40.

Factor Database, Factors of the numbers 4z!+1

Example

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.

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

Prog

(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

Crossrefs

Complement of A152089. A171099 gives the number of solutions.

Cf. A000142, A000217, A051533, A076680.

Sequence in context: A050728 A230999 A380572 * A289342 A286302 A299767

Adjacent sequences: A180587 A180588 A180589 * A180591 A180592 A180593

Keyword

nonn,more

Author

Robert G. Wilson v, Sep 10 2010

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

Status

approved