

A084953


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


3



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
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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, Java Applet: Sum of squares.
Rob Burns, Factorials and Legendre's threesquare theorem, arXiv:2101.01567 [math.NT], 2021.
JeanMarc 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. 243251.


FORMULA

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


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

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


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


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



