OFFSET
1,1
COMMENTS
Crandall & Pomerance refer to these numbers in reference to 2638^4405 + 4405^2638, which was then the largest known prime of this form. - Alonso del Arte, Apr 05 2006 [Comment amended by N. J. A. Sloane, Apr 06 2015]
Conjecture: For d > 11, 10^(d-1)+(d-1)^10 is the smallest (base ten) d-digit term. - Hans Havermann, May 21 2018
Conjecture from Zhi-Wei Sun, Feb 26 2022: (Start)
(i) For each n > 0, we have a(n) <= p+1 < a(n+1) for some prime p.
(ii) a(n) < p < a(n+1) for some prime p, except that the interval (a(5), a(6)) = (54, 57) contains no prime. (End)
A013499 \ {1} is the subsequence of terms of the form 2*n^n, n > 1. - Bernard Schott, Mar 26 2022
REFERENCES
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005.
LINKS
Hans Havermann, Table of n, a(n) for n = 1..5000 (terms 1..1001 from T. D. Noe)
Wikipedia, Leyland number.
EXAMPLE
a(9) = 177 because we can write 177 = 2^7 + 7^2.
MAPLE
N:= 10^7: # to get all terms <= N
A:= {3}:
for n from 2 to floor(N^(1/2)) do
for k from 2 do
a:= n^k + k^n;
if a > N then break fi;
A:= A union {a};
od
od:
A; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A, list)); # Robert Israel, Apr 13 2015
MATHEMATICA
Take[Sort[Flatten[Table[x^y + y^x, {x, 2, 100}, {y, x, 100}]]], 42] (* Alonso del Arte, Apr 05 2006 *)
nn=10^50; n=1; Union[Reap[While[n++; num=2*n^n; num<nn, Sow[num]; k=n; While[k++; num=n^k+k^n; num<nn, Sow[num]]]][[2, 1]]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Oct 23 2002
EXTENSIONS
More terms from Benoit Cloitre, Oct 24 2002
More terms from Alonso del Arte, Apr 05 2006
STATUS
approved