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

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

Offset: 1

Author

Juri-Stepan Gerasimov, Sep 02 2017

Keywords

• nonn
• more

Comments

From Robert Israel, Sep 04 2017: (Start)

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

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

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

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

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

(End)

Examples

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

Links

• Robert Israel, Upper bounds on a(n) for n = 1..2000

Crossrefs

Cf. A285929, A289629.

Cf. A280302, A280547.

Programs

Maple

A289985 := proc(n)
    local k;
    for k from 1 do
        if not issqrfree((n+1)^k+(-n)^k) then
            return k;
        end if;
    end do:
end proc:
for n from 1 do
    printf("%d,\n",A289985(n)) ;
end do: # R. J. Mathar, Sep 04 2017

Mathematica

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

Extensions

a(24)-a(41) from Giovanni Resta, Sep 04 2017