A334213 -id:A334213 - 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))

A334214 Odd numbers m such that m^k + 1 is squarefree for all 0 <= k <= (m - 1)/2.

Original entry on oeis.org

1, 5, 9, 13, 21, 25, 85, 105, 165

Offset: 1

Gionata Neri, Apr 18 2020

a(10) > 10000, if it exists. Most of the relevant factorizations of 165^k+1 were already present in the FactorDB database. The factorization of 165^79+1 has been completed with the software CADO-NFS. - Giovanni Resta, Apr 22 2020

Cf. A334212, A334213.

isOK(m) = k=0; until(k>(m-1)/2, if(!issquarefree(m^k+1), return(0)); k++); 1;
for(m=0, 140, if(isOK(m)&&m%2==1, print1(m, ", ")))

a(9) from Giovanni Resta, Apr 22 2020

cp's OEIS Frontend

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

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