OFFSET
1,1
COMMENTS
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.
There are no primes of this form for n <= 3800. - Michael S. Branicky, Dec 15 2021
REFERENCES
D. E. Knuth, The Art of Computer Programming, Volume 1, 1997, p. 116, problem 7.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..37
FORMULA
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)).
MATHEMATICA
Select[Table[Product[k^k, {k, n}]+11, {n, 10}], CompositeQ] (* Harvey P. Dale, Jun 12 2016 *)
PROG
(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", ")); ) }
(Python)
from sympy import isprime
from itertools import count
def agen():
p = 1
for k in count(1):
p *= k**k
if not isprime(p + 11):
yield p + 11
g = agen()
print([next(g) for n in range(1, 12)]) # Michael S. Branicky, Dec 15 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Feb 12 2003
STATUS
approved