A188047 Numbers k such that k^k-1 and k^k+1 are squarefree.

2, 4, 6, 12, 16, 20, 22, 34, 36, 42, 52, 56, 58, 60, 66, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 138, 140, 142, 144, 156, 158

Offset

1,1

Comments

From Kevin P. Thompson, May 03 2022: (Start)

a(39) >= 160 (160^160-1 is squarefree; 160^160+1 has no known square factors but is not completely factored).

162, 186, 198, and 256 are also terms in this sequence. (End)

Links

Table of n, a(n) for n=1..38.

Status of 160^160+1

Example

6 is a term since 6^6-1 = 46655 = 5*7*31*43 and 6^6+1 = 46657 = 13*37*97 are both squarefree.

Mathematica

Select[Range@42, SquareFreeQ[#^#-1]&&SquareFreeQ[#^#+1]&]

Prog

(PARI) isok(k) = issquarefree(k^k-1) && issquarefree(k^k+1); \\ Michel Marcus, Feb 22 2021

Crossrefs

Intersection of A184966 and A184967.

Cf. A005117.

Sequence in context: A309096 A019280 A090748 * A032465 A089395 A089699

Adjacent sequences: A188044 A188045 A188046 * A188048 A188049 A188050

Keyword

nonn,hard,more

Author

Vladimir Joseph Stephan Orlovsky, Mar 19 2011

Extensions

a(11)-a(25) from D. S. McNeil, Mar 22 2011

a(26)-a(31) from Amiram Eldar, Feb 22 2021

a(32)-a(38) (from FactorDB) added by Kevin P. Thompson, May 03 2022

Status

approved