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 A068869 Smallest number k such that n! + k is a square. 11
 0, 2, 3, 1, 1, 9, 1, 81, 729, 225, 324, 39169, 82944, 176400, 215296, 3444736, 26167684, 114349225, 255004929, 1158920361, 11638526761, 42128246889, 191052974116, 97216010329, 2430400258225, 1553580508516, 4666092737476 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Observation: for n <2000, only for n = 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16 is a(n) a square. According to my conjecture that n! + n^2 != m^2 for n>=1, m>=0 (see A004664), for all terms of A068869 the following will be true: A068869(n) != n^2 [From Alexander R. Povolotsky, Oct 06 2008] There are two cases: a(n) > sqrt(n!) in A182203 and a(n) < sqrt(n!) in A182204. [Artur Jasinski, Apr 13 2012] LINKS T. D. Noe, Table of n, a(n) for n=1..100 FORMULA a(n) = A055228(n)^2-n! = ceiling(sqrt(n!))^2-n! = A048761(n!)-n!. A068869(n) <= A038202(n)^2, with equality for the n listed in the first comment. - M. F. Hasler, Apr 01 2012 EXAMPLE a(6) = 9 as 6! + 9 = 729 is a square. MATHEMATICA Table[ Ceiling[ Sqrt[n! ]]^2 - n!, {n, 1, 28}] PROG (PARI) A068869(n)=(sqrtint(n!-1)+1)^2-n!  \\ M. F. Hasler, Apr 01 2012 CROSSREFS Cf. A066857. Sequence in context: A135900 A173272 A047789 * A251046 A064529 A322874 Adjacent sequences:  A068866 A068867 A068868 * A068870 A068871 A068872 KEYWORD nonn AUTHOR Amarnath Murthy, Mar 13 2002 EXTENSIONS More terms from Vladeta Jovovic, Mar 21 2002 Edited by Robert G. Wilson v and N. J. A. Sloane, Mar 22 2002 STATUS approved

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Last modified June 16 14:38 EDT 2019. Contains 324152 sequences. (Running on oeis4.)