A307244 a(0) = 1; a(n) is the smallest integer k > a(n-1) such that 3^(k-1) == 1 (mod a(n-1)*k).

1, 2, 5, 13, 19, 37, 73, 97, 193, 241, 601, 751, 2251, 3001, 4001, 16001, 96001, 160001, 1120001, 4480001, 13440001, 20160001, 23385601, 29232001, 36540001, 38628001, 115884001, 231768001, 579420001, 1448550001, 1931400001, 2172825001, 6518475001, 22814662501, 53234212501, 425873700001, 1703494800001

Offset: 0

Author

Thomas Ordowski, Mar 30 2019

Keywords

• nonn

Comments

For n > 0, a(n) is prime or pseudoprime (a Fermat pseudoprime to base 3).

Conjecture: a(n) is prime for every n > 0, namely a(n) is the smallest prime p > a(n-1) different from 3 such that 3^(p-1) == 1 (mod a(n-1)), with a(0) = 1.

Generally: for a fixed integer base b > 1, a(n) is the smallest k > a(n-1) such that b^(k-1) == 1 (mod a(n-1)*k), with a(0) = 1. For n > 0, a(n) is prime or pseudoprime (a Fermat pseudoprime to base b). If for a base b, a(n) is a prime for every n > 0, then a(n) is the smallest prime p > a(n-1) that does not divide b such that b^(p-1) == 1 (mod a(n-1)), with a(0) = 1. For any integer base b > 1, a(n) is prime for almost all n. Seems that at most finitely many terms are composite.

Crossrefs

Cf. A306826.

Programs

Mathematica

A = {1}; While[Length[A] < 500, a = Last[A]; r = MultiplicativeOrder[3, a]; k = a + r; While[PowerMod[3, k - 1, k a] != 1, k = k + r];  AppendTo[A, k]]; Take[A, 75] (* Emmanuel Vantieghem, Mar 31 2019 *)

Extensions

a(18)-a(29) from Amiram Eldar, Mar 30 2019

More terms from Emmanuel Vantieghem, Mar 31 2019