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
Keywords
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.
Programs
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C
/* See Alpern link. */
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Mathematica
Select[Range[500], Mod[#!/4^IntegerExponent[#!, 4], 8] == 7 &] (* Amiram Eldar, Jan 11 2021 *)
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PARI
isA004215(n)= n\4^valuation(n, 4)%8==7; isok(n) = isA004215(n!); \\ Michel Marcus, Dec 30 2020
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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
Comments