A124259 Smallest k such that n + n^2 + ... + n^k is not squarefree.

4, 6, 2, 1, 4, 14, 2, 1, 1, 9, 2, 1, 4, 6, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 2, 1, 1, 4, 3, 2, 1, 4, 9, 2, 1, 4, 4, 2, 1, 4, 20, 2, 1, 1, 9, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 5, 2, 1, 4, 2, 1, 1, 4, 25, 2, 1, 4, 4, 2, 1, 4, 2, 1, 1, 4, 7, 2, 1, 1, 4, 2, 1, 4, 6, 2, 1, 2, 1, 2, 1, 4, 9, 2, 1, 2, 1, 1, 1, 4, 20, 2, 1

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..1589

Formula

A124260(n) = Sum_{k=1..a(n)} n^k.

a(A013929(n)) = 1.

Example

n=5: 5 = A005117(4),
5 + 5^2 = 30 = 2*3*5 = A005117(19),
5 + 5^2 + 5^3 = 155 = 5*31 = A005117(95),
5 + 5^2 + 5^3 + 5^4 = 780 = (2^2)*3*5*13 not squarefree,
therefore a(5) = 4 and A124260(5) = 780.

Maple

A124259 := proc(n)
    local k ;
    if n =1 then
        return 4;
    end if;
    for k from 1 do
        if not numtheory[issqrfree](n*(n^k-1)/(n-1)) then
            return  k;
        end if
    end do:
end proc:
seq(A124259(n), n=1..40) ; # R. J. Mathar, Jan 13 2021

Mathematica

a[n_] := Module[{k = 1, s = n}, While[SquareFreeQ[s], k++; s += n^k]; k]; Array[a, 100] (* Amiram Eldar, Dec 26 2020  *)

Prog

(PARI) a(n) = my(k=1); while (issquarefree(sum(i=1, k, n^i)), k++); k; \\ Michel Marcus, Dec 26 2020

Crossrefs

Cf. A005117, A013929, A124260.

Sequence in context: A375366 A373635 A160327 * A214549 A154892 A286155

Adjacent sequences: A124256 A124257 A124258 * A124260 A124261 A124262

Keyword

nonn

Author

Reinhard Zumkeller, Oct 23 2006

Extensions

Data corrected by Amiram Eldar, Dec 26 2020

Status

approved