A084953 Numbers k such that k! is the sum of 4 but no fewer nonzero squares.

10, 12, 24, 25, 48, 49, 54, 60, 78, 91, 96, 97, 107, 114, 120, 121, 142, 151, 167, 170, 172, 180, 192, 193, 212, 222, 226, 238, 240, 241, 246, 252, 270, 279, 301, 307, 309, 318, 327, 333, 344, 345, 357, 360, 361, 367, 375, 379, 384, 385, 403, 405, 421, 424, 425

Offset: 1

Author

Hugo Pfoertner, Jun 15 2003

Keywords

• nonn

Comments

The asymptotic density of this sequence is 1/8 (Deshouillers and Luca, 2010). - Amiram Eldar, Jan 11 2021

Examples

			a(1) = 10 because 10! cannot be written as the sum of fewer than 4 squares.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..5000
• Dario Alpern, Sum of squares web application. See also code on GitHub.
• Rob Burns, Factorials and Legendre's three-square theorem, arXiv:2101.01567 [math.NT], 2021.
• Jean-Marc Deshouillers and Florian Luca, How often is n! a sum of three squares?, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 243-251.

Crossrefs

Cf. A000142, A004215, A084966.

Complement of A267215.

Programs

C

/* See Alpern link. */

Mathematica

Select[Range[500], Mod[#!/4^IntegerExponent[#!, 4], 8] == 7 &] (* Amiram Eldar, Jan 11 2021 *)

PARI

isA004215(n)= n\4^valuation(n, 4)%8==7;
isok(n) = isA004215(n!); \\ Michel Marcus, Dec 30 2020

Python

from math import factorial
from itertools import count, islice
def A084953_gen(startvalue=1): # generator of terms >= startvalue
        return filter(lambda n:(factorial(n)>>((n-n.bit_count())&-2))&7==7,count(max(startvalue,1)))
A084953_list = list(islice(A084953_gen(),30)) # Chai Wah Wu, Jul 09 2022

Formula

Equivalently, k! is of the form (4^i)*(8*j+7), i >= 0, j >= 0.

a(n) ~ 8n. - Charles R Greathouse IV, Jan 06 2025

Extensions

Edited and extended by Robert G. Wilson v, Jun 17 2003

Added missing term 357 by Rob Burns, Dec 30 2020