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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 (list; graph; refs; listen; history; text; internal format)
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 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.

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

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Last modified June 23 21:09 EDT 2021. Contains 345402 sequences. (Running on oeis4.)