A124259 - OEIS

%I #15 Jan 13 2021 04:20:42

%S 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,

%T 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,

%U 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

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

%H Amiram Eldar, <a href="/A124259/b124259.txt">Table of n, a(n) for n = 1..1589</a>

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

%F a(A013929(n)) = 1.

%e n=5: 5 = A005117(4),

%e 5 + 5^2 = 30 = 2*3*5 = A005117(19),

%e 5 + 5^2 + 5^3 = 155 = 5*31 = A005117(95),

%e 5 + 5^2 + 5^3 + 5^4 = 780 = (2^2)*3*5*13 not squarefree,

%e therefore a(5) = 4 and A124260(5) = 780.

%p A124259 := proc(n)

%p     local k ;

%p     if n =1 then

%p         return 4;

%p     end if;

%p     for k from 1 do

%p         if not numtheory[issqrfree](n*(n^k-1)/(n-1)) then

%p             return  k;

%p         end if

%p     end do:

%p end proc:

%p seq(A124259(n),n=1..40) ; # _R. J. Mathar_, Jan 13 2021

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

%o (PARI) a(n) = my(k=1); while (issquarefree(sum(i=1, k, n^i)), k++); k; \\ _Michel Marcus_, Dec 26 2020

%Y Cf. A005117, A013929, A124260.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Oct 23 2006

%E Data corrected by _Amiram Eldar_, Dec 26 2020