A076980 - OEIS

%I #59 Mar 27 2022 13:39:23

%S 3,8,17,32,54,57,100,145,177,320,368,512,593,945,1124,1649,2169,2530,

%T 4240,5392,6250,7073,8361,16580,18785,20412,23401,32993,60049,65792,

%U 69632,93312,94932,131361,178478,262468,268705,397585,423393,524649,533169

%N 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)).

%C 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]

%C Conjecture: For d > 11, 10^(d-1)+(d-1)^10 is the smallest (base ten) d-digit term. - _Hans Havermann_, May 21 2018

%C Conjecture from _Zhi-Wei Sun_, Feb 26 2022: (Start)

%C (i) For each n > 0, we have a(n) <= p+1 < a(n+1) for some prime p.

%C (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)

%C A013499 \ {1} is the subsequence of terms of the form 2*n^n, n > 1. - _Bernard Schott_, Mar 26 2022

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

%H Hans Havermann, <a href="/A076980/b076980.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..1001 from T. D. Noe)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Leyland_number">Leyland number</a>.

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

%p N:= 10^7: # to get all terms <= N

%p A:= {3}:

%p for n from 2 to floor(N^(1/2)) do

%p   for k from 2 do

%p      a:= n^k + k^n;

%p      if a > N then break fi;

%p      A:= A union {a};

%p   od

%p od:

%p A; # if using Maple 11 or earlier, uncomment the next line

%p # sort(convert(A,list)); # _Robert Israel_, Apr 13 2015

%t Take[Sort[Flatten[Table[x^y + y^x, {x, 2, 100}, {y, x, 100}]]], 42] (* _Alonso del Arte_, Apr 05 2006 *)

%t 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]]]

%Y Prime subset of this sequence, A094133.

%Y Cf. A013499.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Oct 23 2002

%E More terms from _Benoit Cloitre_, Oct 24 2002

%E More terms from _Alonso del Arte_, Apr 05 2006