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

%I #17 May 04 2022 07:00:57

%S 2,4,6,12,16,20,22,34,36,42,52,56,58,60,66,72,78,84,86,88,90,92,94,96,

%T 102,104,106,108,110,112,114,128,138,140,142,144,156,158

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

%C From _Kevin P. Thompson_, May 03 2022: (Start)

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

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

%H <a href="http://factordb.com/index.php?id=1100000000009541993">Status of 160^160+1</a>

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

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

%o (PARI) isok(k) = issquarefree(k^k-1) && issquarefree(k^k+1); \\ _Michel Marcus_, Feb 22 2021

%Y Intersection of A184966 and A184967.

%Y Cf. A005117.

%K nonn,hard,more

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Mar 19 2011

%E a(11)-a(25) from _D. S. McNeil_, Mar 22 2011

%E a(26)-a(31) from _Amiram Eldar_, Feb 22 2021

%E a(32)-a(38) (from FactorDB) added by _Kevin P. Thompson_, May 03 2022