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

%I A242872 #16 May 15 2023 08:44:42
%S A242872 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,
%T A242872 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,
%U A242872 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
%N A242872 Least number k > 1 such that (k^k-n^n)/(k-n) is an integer.
%C A242872 a(n) <= n-1 for n > 2 (since k > 1).
%C A242872 This is also the least number k such that (k^n-n^k)/(k-n) is an integer.
%H A242872 Robert Israel, <a href="/A242872/b242872.txt">Table of n, a(n) for n = 1..10000</a>
%e A242872 (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.
%p A242872 A242872:= proc(n)
%p A242872    local nn, k;
%p A242872    nn:= n^n;
%p A242872    for k from 2 to n-1 do
%p A242872       if (nn-k^k) mod (n-k) = 0 then return k fi
%p A242872    od;
%p A242872    return n+1;
%p A242872 end:
%p A242872 seq(A242872(n),n=1..100); # _Robert Israel_, May 25 2014
%t A242872 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]]]]];
%t A242872 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, May 15 2023 *)
%o A242872 (PARI) a(n)=for(k=2,n+1,if(k!=n,s=(k^k-n^n)/(k-n);if(floor(s)==s,return(k))));
%o A242872 n=1;while(n<100,print(a(n));n+=1)
%Y A242872 Cf. A242870, A242871.
%K A242872 nonn
%O A242872 1,1
%A A242872 _Derek Orr_, May 24 2014