A289985 - OEIS

%I #25 Dec 07 2021 11:00:11

%S 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 34,2,21,78,10,68,10,41,57,10,55,10,55,21,10

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

%C From _Robert Israel_, Sep 04 2017: (Start)

%C If (n+1)^k + (-n)^k is divisible by p^2 then so is (m+1)^k + (-m)^k

%C for m == n (mod p^2), so a(m) <= k for such m.

%C For example, a(n) = 2 if n == 3 or 21 (mod 25).

%C a(24) = 78 because 25^78 + (-24)^78 is divisible by 13^2.

%C a(42) <= 171 because 43^171 + (-42)^171 is divisible by 19^2.

%C (End)

%H Robert Israel, <a href="/A289985/a289985.txt">Upper bounds on a(n) for n = 1..2000</a>

%e a(1) = 10 because (1+1)^10 + (-1)^10 = 1025 is divisible by 5^2.

%p A289985 := proc(n)

%p     local k;

%p     for k from 1 do

%p         if not issqrfree((n+1)^k+(-n)^k) then

%p             return k;

%p         end if;

%p     end do:

%p end proc:

%p for n from 1 do

%p     printf("%d,\n",A289985(n)) ;

%p end do: # _R. J. Mathar_, Sep 04 2017

%t Table[SelectFirst[Range[10^2], ! SquareFreeQ[(n + 1)^# + (-n)^#] &], {n, 23}] (* _Michael De Vlieger_, Sep 03 2017 *)

%Y Cf. A285929, A289629.

%Y Cf. A280302, A280547.

%K nonn,more

%O 1,1

%A _Juri-Stepan Gerasimov_, Sep 02 2017

%E a(24)-a(41) from _Giovanni Resta_, Sep 04 2017