A013499 -id:A013499 - cp's OEIS frontend

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

Amarnath Murthy, Oct 23 2002

nonn

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

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

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005.

Hans Havermann, Table of n, a(n) for n = 1..5000 (terms 1..1001 from T. D. Noe)
Wikipedia, Leyland number.

Prime subset of this sequence, A094133.

Cf. A013499.

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

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

More terms from Benoit Cloitre, Oct 24 2002

More terms from Alonso del Arte, Apr 05 2006

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073

Offset: 0

Henry Bottomley, Jun 08 2000

Columns and rows are A000012 (apart from first term), A000027, A001580, A001585, A001589, A001593 etc. Diagonals include A013499 (apart from first two terms), A051442, A051489.
Cf. A055651.
Contribution from Franklin T. Adams-Watters, Oct 26 2009: (Start)
Rows/columns: A000027, A001580, A001585, A001589, A001593, A001594, A001596.
Main diagonal is 2 * A000312. More diagonals: A051442, A051489, A155539.
Cf. A076980, A156353, A156354. (End)

E.g.f. Sum(n,m, T(n,m)/(n! m!)) = e^(x e^y) + e^(y e^x). [From Franklin T. Adams-Watters, Oct 26 2009]

A062207 a(n) = 2*n^n-1.

Original entry on oeis.org

1, 1, 7, 53, 511, 6249, 93311, 1647085, 33554431, 774840977, 19999999999, 570623341221, 17832200896511, 605750213184505, 22224013651116031, 875787780761718749, 36893488147419103231, 1654480523772673528353, 78692816150593075150847, 3956839311320627178247957

Offset: 0

Jason Earls, Jun 13 2001

Also: a(n) = 2m-1 where m is given by Sum_{i = 1..m } 2*i-1 = n^(2*n) (A062206).

"By setting n=m^p, one sees that m^(2p), an even power of any integer, is equal to the sum of all the odd integers up to and including 2m^p-1;..." - p. 16.

			a(2)=7 and 1+3+5+7=16, which is A062206(2).
a(3)=53 and 1+3+5+...+53=729, which is A062206(3).

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 16-17.

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A013499, A062206.

Table[2n^n-1,{n,20}] (* Harvey P. Dale, Jul 19 2015 *)

{ for (n=1, 100, write("b062207.txt", n, " ", 2*(n^n) - 1) ) } \\ Harry J. Smith, Aug 02 2009

a(n) = A013499(n) - 1 for n>=2. - R. J. Mathar, May 18 2007

E.g.f.: 2/(1 + LambertW(-x)) - exp(x). - Vaclav Kotesovec, Dec 21 2014

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001
Definition simplified by M. F. Hasler, Sep 02 2012
a(0)=1 prepended by Alois P. Heinz, Feb 20 2023

cp's OEIS Frontend

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)).

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A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

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A062207 a(n) = 2*n^n-1.

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