A076980 Leyland numbers: 3, together with numbers expressible as n^k + k^n nontrivially, i.e., n,k > 1 (to avoid n = (n-1)^1 + 1^(n-1)).

3, 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 93312, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169, 1048976

Offset: 1

Author

Amarnath Murthy, Oct 23 2002

Keywords

• nonn

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

Examples

			a(9) = 177 because we can write 177 = 2^7 + 7^2.
		

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.

Crossrefs

Prime subset of this sequence, A094133.

Cf. A013499.

Programs

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
				

Extensions

More terms from Benoit Cloitre, Oct 24 2002

More terms from Alonso del Arte, Apr 05 2006