A273618 Numbers m = 2*k+1 where k is odd with the property that 3^k mod m = 1 and k^k mod m = 1.

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

Author

Alzhekeyev Ascar M, May 26 2016

Keywords

• nonn

Comments

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.

The subsequence of these composites begins: 143193768587, 440097066011, 1188059560451, 1392770336147, 1640446291859, 2526966350771, 3639120872171, 3989703695867, 4202422108523, ....

Perhaps this sequence contains all the terms of the sequence A107007 (except 3) or A168539.

Examples

			m=131; 131=2*65+1; 3^65 mod 131 = 1 and 65^65 mod 131 = 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A176997.

Cf. A001262, A001567, A107007, A168539, A214151.

Programs

Maple

filter:= proc(n) local k;
  k:= (n-1)/2;
  3 &^ k mod n = 1 and k &^ k mod n = 1
end proc:
select(filter, [seq(i,i=3..3000, 4)]); # Robert Israel, Nov 28 2019

Mathematica

2#+1&/@Select[Range[1,1200,2],PowerMod[3,#,2#+1]==PowerMod[ #,#,2#+1] == 1&] (* Harvey P. Dale, May 05 2022 *)