A181896 - OEIS

%I #25 Jan 28 2023 11:51:16

%S 5,11,27,71,201,603,1905,6318,21888,78912,295260,1143536,4574144,

%T 18859680,80014848,348776640,1559776320,7147792848,33526120320,

%U 160785625902,787685472000,3938427360000,20082117976800,104349745817240,552166953609600,2973510046027938

%N Least value x solving x^2 - y^2 = n!

%C Many of terms in this sequence are that same as A055228(n) but not all.

%C a(n) solves the Brocard-Ramanujan Problem, n! = a(n)^2 - 1, and thus (n, a(n)) are a pair of Brown Numbers, if and only if A038202(n) = 1. - _Austin Hinkel_, Dec 28 2022

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BrocardsProblem.html">Brocard's Problem</a>.

%t cc = {}; Do[f = n!/4; x = Max[Select[Divisors[f], # <= Sqrt[f] &]]; kk = f/x - x; AppendTo[cc, Sqrt[n! + kk^2]], {n, 4, 30}]; cc

%o (PARI) a(n)=my(N=n!,x=sqrtint(N));while(!issquare(x++^2-N),);x \\ _Charles R Greathouse IV_, Apr 10 2012

%Y For least y value see A038202.

%Y Cf. A055228.

%K nonn

%O 4,1

%A _Artur Jasinski_, Mar 31 2012