A242872 Least number k > 1 such that (k^k-n^n)/(k-n) is an integer.

2, 3, 2, 2, 3, 2, 3, 2, 3, 4, 3, 3, 4, 2, 3, 4, 5, 6, 3, 2, 3, 2, 3, 4, 4, 6, 3, 4, 5, 3, 4, 8, 6, 4, 3, 4, 5, 2, 3, 4, 5, 3, 3, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 4, 5, 2, 3, 4, 5, 5, 7, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 9, 10, 3, 4, 5, 6, 7, 8, 3, 4, 3, 4, 4, 2, 3, 4, 5, 6, 7, 8, 9, 4

Offset: 1

Author

Derek Orr, May 24 2014

Keywords

• nonn

Comments

a(n) <= n-1 for n > 2 (since k > 1).

This is also the least number k such that (k^n-n^k)/(k-n) is an integer.

Examples

			(2^2-5^5)/(2-5) = 3121/3 is not an integer. (3^3-5^5)/(3-5) = 3098/2 = 1549 is an integer. Thus a(5) = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A242870, A242871.

Programs

Maple

A242872:= proc(n)
   local nn, k;
   nn:= n^n;
   for k from 2 to n-1 do
      if (nn-k^k) mod (n-k) = 0 then return k fi
   od;
   return n+1;
end:
seq(A242872(n),n=1..100); # Robert Israel, May 25 2014

Mathematica

a[n_] := Switch[n, 1, 2, 2, 3, _, With[{nn = n^n}, For[k = 2, True, k++, If[Mod[nn-k^k, n-k] == 0, Return[k]]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 15 2023 *)

PARI

a(n)=for(k=2,n+1,if(k!=n,s=(k^k-n^n)/(k-n);if(floor(s)==s,return(k))));
n=1;while(n<100,print(a(n));n+=1)