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

Author

Vladimir Joseph Stephan Orlovsky, Mar 19 2011

Keywords

• nonn
• hard
• more

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)

Examples

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

Links

• Status of 160^160+1

Crossrefs

Intersection of A184966 and A184967.

Cf. A005117.

Programs

Mathematica

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

PARI

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

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