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A289985 Smallest positive k such that (n+1)^k + (-n)^k is divisible by a square greater than 1.

%I A289985 #25 Dec 07 2021 11:00:11
%S A289985 10,11,2,55,21,10,3,10,33,26,10,21,10,5,21,10,55,10,8,2,2,3,7,78,55,3,
%T A289985 34,2,21,78,10,68,10,41,57,10,55,10,55,21,10
%N A289985 Smallest positive k such that (n+1)^k + (-n)^k is divisible by a square greater than 1.
%C A289985 From _Robert Israel_, Sep 04 2017: (Start)
%C A289985 If (n+1)^k + (-n)^k is divisible by p^2 then so is (m+1)^k + (-m)^k
%C A289985 for m == n (mod p^2), so a(m) <= k for such m.
%C A289985 For example, a(n) = 2 if n == 3 or 21 (mod 25).
%C A289985 a(24) = 78 because 25^78 + (-24)^78 is divisible by 13^2.
%C A289985 a(42) <= 171 because 43^171 + (-42)^171 is divisible by 19^2.
%C A289985 (End)
%H A289985 Robert Israel, <a href="/A289985/a289985.txt">Upper bounds on a(n) for n = 1..2000</a>
%e A289985 a(1) = 10 because (1+1)^10 + (-1)^10 = 1025 is divisible by 5^2.
%p A289985 A289985 := proc(n)
%p A289985     local k;
%p A289985     for k from 1 do
%p A289985         if not issqrfree((n+1)^k+(-n)^k) then
%p A289985             return k;
%p A289985         end if;
%p A289985     end do:
%p A289985 end proc:
%p A289985 for n from 1 do
%p A289985     printf("%d,\n",A289985(n)) ;
%p A289985 end do: # _R. J. Mathar_, Sep 04 2017
%t A289985 Table[SelectFirst[Range[10^2], ! SquareFreeQ[(n + 1)^# + (-n)^#] &], {n, 23}] (* _Michael De Vlieger_, Sep 03 2017 *)
%Y A289985 Cf. A285929, A289629.
%Y A289985 Cf. A280302, A280547.
%K A289985 nonn,more
%O A289985 1,1
%A A289985 _Juri-Stepan Gerasimov_, Sep 02 2017
%E A289985 a(24)-a(41) from _Giovanni Resta_, Sep 04 2017