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%I #41 Jul 08 2023 17:06:17
%S 10,12,24,25,48,49,54,60,78,91,96,97,107,114,120,121,142,151,167,170,
%T 172,180,192,193,212,222,226,238,240,241,246,252,270,279,301,307,309,
%U 318,327,333,344,345,357,360,361,367,375,379,384,385,403,405,421,424,425
%N Numbers k such that k! is the sum of 4 but no fewer nonzero squares.
%C The asymptotic density of this sequence is 1/8 (Deshouillers and Luca, 2010). - _Amiram Eldar_, Jan 11 2021
%H Hugo Pfoertner, <a href="/A084953/b084953.txt">Table of n, a(n) for n = 1..5000</a>
%H Dario Alpern, <a href="https://www.alpertron.com.ar/FSQUARES.HTM">Sum of squares web application</a>.
%H Rob Burns, <a href="https://arxiv.org/abs/2101.01567">Factorials and Legendre's three-square theorem</a>, arXiv:2101.01567 [math.NT], 2021.
%H Jean-Marc Deshouillers and Florian Luca, <a href="https://doi.org/10.1007/978-1-4419-6263-8_14">How often is n! a sum of three squares?</a>, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 243-251.
%F Equivalently, k! is of the form (4^i)*(8*j+7), i >= 0, j >= 0.
%e a(1) = 10 because 10! cannot be written as the sum of fewer than 4 squares.
%t Select[Range[500], Mod[#!/4^IntegerExponent[#!, 4], 8] == 7 &] (* _Amiram Eldar_, Jan 11 2021 *)
%o See link.
%o (PARI) isA004215(n)= n\4^valuation(n, 4)%8==7;
%o isok(n) = isA004215(n!); \\ _Michel Marcus_, Dec 30 2020
%o (Python 3.10+)
%o from math import factorial
%o from itertools import count, islice
%o def A084953_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:(factorial(n)>>((n-n.bit_count())&-2))&7==7,count(max(startvalue,1)))
%o A084953_list = list(islice(A084953_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%Y Cf. A000142, A004215, A084966.
%Y Complement of A267215.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Jun 15 2003
%E Edited and extended by _Robert G. Wilson v_, Jun 17 2003
%E Added missing term 357 by _Rob Burns_, Dec 30 2020
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