A359013 Numbers k that can be written as the sum of a perfect square and a factorial in exactly 3 distinct ways.

145, 46249, 63121, 42916624, 18700677890064, 28112213204100, 41654823930457982576640000, 445860623276908458083942400, 666474080134036599385635225600

Offset: 1

Author

Walter Robinson, Dec 11 2022

Keywords

• nonn
• hard
• more

Comments

This does not count x^2 and (-x)^2 as distinct, nor does it count 0! and 1! as distinct.

a(10) > 10^30 if it exists. - David A. Corneth, Dec 11 2022

Examples

			145 = 5^2 + 5! = 11^2 + 4! = 12^2 + 1!.
		

Crossrefs

Cf. A000142, A000290.

Subset of A358071.

Programs

Python

import math
for x in range(1, 120000000):
    total = 0
    prod = 1
    factInc = 2
    while prod <= x:
        sq = math.sqrt(x - prod)
        if sq % 1 == 0:
            total = total + 1
        prod = prod * factInc
        factInc = factInc + 1
    if total == 3:
        print(x)

Extensions

a(5)-a(9) from David A. Corneth, Dec 11 2022