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%I #11 Jul 12 2015 15:51:05
%S 27649,86400001,4031078400001,3319766398771200001,
%T 55696437941726556979200001,21577941222941856209168026828800001,
%U 215779412229418562091680268288000000000000001
%N Composite numbers of the form 1^1 * 2^2 * 3^3 * 4^4 * ... * n^n + 1.
%C No primes other than 2,5,109 found in this sequence for n <= 1000. Conjecture: There are no primes in the sequence 2^2 * 3^3 * 4^4 * ... *n^n + 1 for n > 3. Conjecture: There are no primes in the sequence 2^2 * 3^3 * 4^4 * ... * n^n + 61 for all n.
%D D. E. Knuth, The Art of Computer Programming, Volume 1, 1997, p. 116, problem 7.
%F Prod(k^k, k=1..n)+1 is Composite. Exp(log(1) + 2log(2) + 3log(3) + ... klog(k)) = exp(Sum(k*log(k), k=1..n)).
%t Select[1+#&/@FoldList[Times,1,Table[n^n,{n,10}]],!PrimeQ[#]&] (* _Harvey P. Dale_, May 02 2011 *)
%o (PARI) pcomposits(n,b) = { for(x=1,n, p=1; for(y=1,x, p = p*(y^y); ); if(!isprime(p+b),print1(p+b",")); ) }
%Y Cf. A002109.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Feb 12 2003