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Composite numbers of the form 1^1 * 2^2 * 3^3 * 4^4 * ... * n^n + 11.
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%I #13 Dec 15 2021 07:50:25

%S 12,15,119,27659,86400011,4031078400011,3319766398771200011,

%T 55696437941726556979200011,21577941222941856209168026828800011,

%U 215779412229418562091680268288000000000000011

%N Composite numbers of the form 1^1 * 2^2 * 3^3 * 4^4 * ... * n^n + 11.

%C No primes of this form in the sequence for n <= 60. 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 + 11 for all n. Conjecture: There are no primes in the sequence 2^2 * 3^3 * 4^4 * ... * n^n + 61 for all n.

%C There are no primes of this form for n <= 3800. - _Michael S. Branicky_, Dec 15 2021

%D D. E. Knuth, The Art of Computer Programming, Volume 1, 1997, p. 116, problem 7.

%H Michael S. Branicky, <a href="/A079322/b079322.txt">Table of n, a(n) for n = 1..37</a>

%F Prod(k^k, k=1..n) + 11 is composite. Exp(log(1) + 2log(2) + 3log(3) + ... klog(k)) = exp(Sum(k*log(k), k=1..n)).

%t Select[Table[Product[k^k,{k,n}]+11,{n,10}],CompositeQ] (* _Harvey P. Dale_, Jun 12 2016 *)

%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",")); ) }

%o (Python)

%o from sympy import isprime

%o from itertools import count

%o def agen():

%o p = 1

%o for k in count(1):

%o p *= k**k

%o if not isprime(p + 11):

%o yield p + 11

%o g = agen()

%o print([next(g) for n in range(1, 12)]) # _Michael S. Branicky_, Dec 15 2021

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Feb 12 2003