A394730 Numbers k such that k! is the sum of a square and a triangular number.

0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 15, 19, 23, 25, 27, 31, 35, 43, 46, 47, 48, 53, 54, 57, 58, 60, 69, 71, 78, 83

Offset

1,3

Links

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

Formula

k is a term iff each prime factor congruent to 5 or 7 (mod 8) of 8*k!+1 comes in even degree.

Prog

(Python)
from math import factorial
from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_quadratic
from sympy.abc import x, y
def A394730_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda k:bool(len(diop_quadratic(2*(factorial(k)-x**2)-y*(y+1)))), count(max(startvalue, 0)))
A394730_list = list(islice(A394730_gen(), 20)) # Chai Wah Wu, Mar 31 2026

Crossrefs

Cf. A000142, A014133, A230363.

Sequence in context: A096304 A119955 A158573 * A194398 A047603 A047362

Adjacent sequences: A394727 A394728 A394729 * A394731 A394732 A394733

Keyword

nonn,more

Author

Max Alekseyev, Mar 30 2026

Status

approved