

A068869


Smallest number k such that n! + k is a square.


12



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
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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)^2n! = ceiling(sqrt(n!))^2n! = 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)^2n! \\ M. F. Hasler, Apr 01 2012


CROSSREFS

Cf. A066857.
Sequence in context: A173272 A326303 A047789 * A251046 A064529 A338263
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



