A334214 -id:A334214 - cp's OEIS frontend

A334212 Least number k such that n^k + 1 is not squarefree.

3, 1, 5, 3, 7, 1, 1, 5, 11, 1, 10, 7, 3, 1, 17, 1, 2, 1, 3, 11, 10, 1, 1, 13, 1, 1, 10, 3, 31, 1, 2, 10, 5, 1, 37, 10, 2, 1, 5, 2, 10, 1, 1, 21, 47, 1, 1, 1, 3, 1, 10, 1, 5, 1, 3, 2, 10, 1, 14, 21, 1, 1, 5, 3, 21, 1, 2, 3, 2, 1, 10, 10, 1, 1, 7, 3, 10, 1

Offset: 2

Gionata Neri, Apr 18 2020

nonn

For n == 1 (mod 4) (n not 1), a(n) <= (n + 1)/2.
For n == 3 (mod 4), a(n) = 1.
For even n, a(n) <= n + 1.
Existence proof for n >= 2 and upper bounds use the binomial formula.

Cf. A013929, A334213, A334214.

for(n=2,79, for(k=1,n+1, !issquarefree(n^k+1)&!print1(k", ")&break))

A334213 Numbers m such that m^k + 1 is squarefree for all 0 <= k <= m.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 16, 30, 36, 46, 256

Offset: 1

Gionata Neri, Apr 18 2020

m = 2^i is a term iff k*i is not in A049096 with 0 < k < m + 1. Up to i = 128, there are no more terms of the form 2^i. a(12) > 10^7, if it exists. - Jinyuan Wang, May 01 2020

			4^0 + 1 = 2 is squarefree, 4^1 + 1 = 5 is squarefree, 4^2 + 1 = 17 is squarefree, 4^3 + 1 = 5*13 is squarefree and 4^4 + 1 = 257 is squarefree, so 4 is in the sequence.

Cf. A049096, A334212, A334214.

Do[L=Length[a];a=Select[a=m^Range[0,m-1]+1,SquareFreeQ[#]&];If[L==m-1,Print[m-1]],{m,0,1000}] (* Metin Sariyar, Apr 21 2020 *)

isOK(m) = k=0; until(k>m, if(!issquarefree(m^k+1), return(0)); k++); 1;
for(m=0, 99, if(isOK(m), print1(m, ", ")))

a(11) from Jinyuan Wang, May 01 2020

cp's OEIS Frontend

A334212 Least number k such that n^k + 1 is not squarefree.

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