A230363 Factorials representable as b^2 + triangular(c).

1, 1, 2, 6, 24, 120, 362880, 3628800, 39916800, 479001600, 6227020800, 1307674368000, 121645100408832000, 25852016738884976640000, 15511210043330985984000000, 10888869450418352160768000000, 8222838654177922817725562880000000, 10333147966386144929666651337523200000000

Offset

1,3

Comments

Numbers k such that k! is representable as a sum of a square and a triangular number are listed in A394730.

k! is a term iff each prime factor congruent to 5 or 7 (mod 8) of 8*k!+1 comes in even degree. - Max Alekseyev, Mar 30 2026

1! = 1/2*1*(1+1), 3! = 1/2*3*(3+1) and 5! = 1/2*15*(15+1)/2 are triangular terms of the sequence. Next such term, if it exists is greater than 10000!. - Farideh Firoozbakht, Oct 18 2013

Links

Max Alekseyev, Table of n, a(n) for n = 1..31

Formula

A014133 INTERSECT A000142. - R. J. Mathar, Oct 11 2014

a(n) = A394730(n)!. - Max Alekseyev, Mar 30 2026

Example

13! = 66708^2+(1/2)*59616*(59616+1) = 78693^2+(1/2)*8298*(8298+1), so 13! = 6227020800 is in the sequence. What is the next term of the sequence which has more than one representation of the form b^2 + triangular(c)? - Farideh Firoozbakht, Oct 18 2013

Prog

(Python)
import math
f=1
for n in range(1, 1000000):
    f *= n
    t = b = 0
    while t<=f:
        x = f-t
        a = math.isqrt(x)
        if a*a==x:
            print(f, end=", ")
            break
        b += 1
        t = b*(b+1)//2

Crossrefs

Cf. A000142, A000290, A000217, A014133, A394730.

Sequence in context: A114779 A383920 A317681 * A358500 A358494 A357922

Adjacent sequences: A230360 A230361 A230362 * A230364 A230365 A230366

Keyword

nonn

Author

Alex Ratushnyak, Oct 17 2013

Extensions

Initial 1 added by Farideh Firoozbakht, Oct 18 2013

Terms a(14) onward from Max Alekseyev, Mar 30 2026

Status

approved