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%I #50 Jul 28 2022 15:18:59
%S 1,1,2,3,5,11,27,71,201,603,1905,6318,21887,78912,295260,1143536,
%T 4574144,18859678,80014835,348776577,1559776269,7147792819,
%U 33526120083,160785623546,787685471323,3938427356615,20082117944246,104349745809074,552166953567229
%N a(n) = ceiling(sqrt(n!)).
%C Axenovich's improvement to the Erdős strong Delta-system conjecture. Erdős and Rado called a family of sets {A1, A2, .., Ak} a strong Delta-system if all the intersections Ai INTERSECT Aj are identical, 1 <= i < j <= k. Denoting by f(n,k) the smallest integer m for which every family of n-sets {A1, A2, .., Am} contains k sets forming a strong Delta-system. Then Axenovich et al. proved f(n,3) < (n!)^((1/2) + epsilon)) < a(n) holds for every epsilon > 0, provided n is sufficiently large. - _Jonathan Vos Post_, Apr 29 2007; typos fixed by _Li-yao Xia_, May 06 2014
%C For n>0, a(n) is the least m>0 such that n! <= m^2. - _Clark Kimberling_, Jul 18 2012
%H Chai Wah Wu, <a href="/A055228/b055228.txt">Table of n, a(n) for n = 0..807</a> (n = 0..300 from T. D. Noe)
%H M. Axenovich, D. Fon-Der-Flaass and A. Kostochka, <a href="http://dx.doi.org/10.1016/0012-365X(94)00185-L">On set systems without weak 3-Delta-subsystems</a>, Discrete Math. 138(1995), 57-62.
%H Bela Bollobas, <a href="http://www.jstor.org/stable/2589077">To Prove and Conjecture: Paul Erdős and His Mathematics</a>, Am. Math. Monthly, 105 (March 1998)3, p. 232.
%H P. Erdős and R. Rado, <a href="http://www.renyi.hu/~p_erdos/1961-07.pdf">Intersection theorems for systems of finite sets I</a>, J. London Math. Soc. (2) 35(1960), 85-90.
%H P. Erdős and R. Rado, <a href="http://www.renyi.hu/~p_erdos/1969-02.pdf">Intersection theorems for systems of finite sets II</a>, J. London Math. Soc. (2) 44(1969), 467-479.
%F a(n) = A003059(A000142(n)). - _Jonathan Vos Post_, Apr 29 2007
%p A055228:=n->ceil(sqrt(n!)); seq(A055228(n), n=0..30); # _Wesley Ivan Hurt_, May 06 2014
%t Table[Ceiling[Sqrt[n!]], {n, 0, 30}] (* _Wesley Ivan Hurt_, May 06 2014 *)
%o (PARI) a(n) = ceil(sqrt(n!)) \\ _Michel Marcus_, Jul 30 2013
%o (Python)
%o from math import isqrt, factorial
%o def A055228(n): return 1+isqrt(factorial(n)-1) # _Chai Wah Wu_, Jul 28 2022
%Y Cf. A000142, A003059.
%K easy,nonn
%O 0,3
%A _Henry Bottomley_, Jun 21 2000
%E A comment stating that one of the terms was wrong has been deleted - the terms are correct. - _T. D. Noe_, Apr 22 2009
%E More terms from _Wesley Ivan Hurt_, May 06 2014
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