%I #45 Aug 21 2019 11:29:38

%S 2,2,5,37,577,14401,518401,25401601,1625702401,131681894401,

%T 13168189440001,1593350922240001,229442532802560001,

%U 38775788043632640001,7600054456551997440001,1710012252724199424000001,437763136697395052544000001

%N a(n) = (n!)^2 + 1.

%C Used to prove there are infinitely many primes of the form 4k+1 (see A282706). - _N. J. A. Sloane_, Feb 26 2017

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.

%D F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.

%D H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.

%D M. Le, On the Interesting Smarandache Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 133-134.

%D M. Le, The Primes in Smarandache Power Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 96-97.

%H G. C. Greubel, <a href="/A020549/b020549.txt">Table of n, a(n) for n = 0..250</a>

%H M. Fleuren, <a href="http://www.gallup.unm.edu/~smarandache/SmSquProd.txt">Smarandache Square Products</a>.

%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">Sequences of Numbers Involved in Unsolved Problems</a>.

%H Apoloniusz Tyszka, <a href="https://doi.org/10.13140/RG.2.2.28996.68486">On sets X, subset of N, whose finiteness implies that we know an algorithm which for every n, element of N, decides the inequality max (X) < n</a>, (2019).

%H Apoloniusz Tyszka, <a href="https://philarchive.org/archive/TYSDASv56">On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheSequences.html">Smarandache Sequences</a>

%p with(combinat):seq(fibonacci(3,n!), n=0..16); # _Zerinvary Lajos_, Apr 21 2008

%p [seq(n!^2+1,n=0..20)]; # _N. J. A. Sloane_, Feb 26 2017

%t Table[(n!)^2 + 1, {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 08 2011 *)

%o (PARI) a(n)=n!^2 + 1 \\ _Charles R Greathouse IV_, Nov 30 2016

%Y Cf. A001044.

%Y For smallest prime factor see A282706.

%K nonn

%O 0,1

%A _N. J. A. Sloane_, _Simon Plouffe_