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

1,1

Comments

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

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.

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

Example

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

Mathematica

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

Prog

(C) /* See Alpern link. */
(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

Crossrefs

Cf. A000142, A004215, A084966.

Complement of A267215.

Sequence in context: A108703 A098785 A022324 * A235686 A087697 A241177

Adjacent sequences: A084950 A084951 A084952 * A084954 A084955 A084956

Keyword

nonn,changed

Author

Hugo Pfoertner, Jun 15 2003

Extensions

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

Added missing term 357 by Rob Burns, Dec 30 2020

Status

approved