A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

91, 7381, 597871, 48427561, 3922632451, 317733228541, 25736391511831, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset

1,1

Comments

These numbers were obtained for values of k from 1 to 20, with the following exceptions: k = 10, 12, 13, 16, 17, 19, for which were obtained 3^n mod n = 3^7, 3^31, 3^37, 3^25, 3^31, 3^13.

Conjecture: There are infinitely many Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8, where k is a natural number.

It is true: for example, when 2k+1 is a prime number (see A210461). - Bruno Berselli, Jan 22 2013

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..190

Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Mathematica

Select[Table[(3^(4k + 2) - 1)/8, {k, 80}], PowerMod[3, # - 1, #] == 1 &] (* Alonso del Arte, May 14 2019 *)

Prog

(PARI) list(lim)=my(v=List(), t); lim\=1; for(k=1, (logint(8*lim+1, 3)-2)\4, t=3^(4*k + 2)>>3; if(Mod(3, t)^t==3, listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

Crossrefs

Cf. A005935, A210461 (subsequence), A217841.

Sequence in context: A234123 A103855 A238541 * A210461 A022253 A172174

Adjacent sequences: A217850 A217851 A217852 * A217854 A217855 A217856

Keyword

nonn

Author

Marius Coman, Oct 12 2012

Status

approved