A079322 Composite numbers of the form 1^1 * 2^2 * 3^3 * 4^4 * ... * n^n + 11.

12, 15, 119, 27659, 86400011, 4031078400011, 3319766398771200011, 55696437941726556979200011, 21577941222941856209168026828800011, 215779412229418562091680268288000000000000011

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

Sequence in context: A024875 A152190 A330367 * A167304 A191966 A389559

Adjacent sequences: A079319 A079320 A079321 * A079323 A079324 A079325

Keyword

easy,nonn

Author

Cino Hilliard, Feb 12 2003

Status

approved