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

1,1

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)

Links

Table of n, a(n) for n=1..41.

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

Example

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

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 *)

Crossrefs

Cf. A285929, A289629.

Cf. A280302, A280547.

Sequence in context: A248025 A303501 A368347 * A065517 A235600 A059514

Adjacent sequences: A289982 A289983 A289984 * A289986 A289987 A289988

Keyword

nonn,more

Author

Juri-Stepan Gerasimov, Sep 02 2017

Extensions

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

Status

approved