A321790 a(n) is the smallest base a > 2 such that a^(k-1) != 1 (mod k), where k = A001567(n), the n-th Fermat pseudoprime to base 2.

3, 3, 3, 5, 3, 7, 3, 3, 5, 5, 7, 3, 3, 3, 3, 3, 3, 7, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 13, 3, 3, 3, 3, 5, 3, 3, 3, 3, 7, 3, 3, 13, 5, 3, 7, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 11, 3, 5, 5, 3, 3, 3, 5, 5, 3, 5, 7, 5, 5, 3, 13, 3, 3

Offset: 1

Author

Thomas Ordowski, Nov 19 2018

Keywords

• nonn

Comments

a(n) <= A177415(n).

Each a(n) is an odd prime.

If k = A001567(n) is a Carmichael number, then a(n) = lpf(k).

Conjecture: if k = A001567(n) is semiprime, then a(n) < lpf(k).

The smallest numbers k = A001567(n) such that a(n) = prime(m) for m > 1 are 341, 1105, 1729, 75361, 29341, 162401, 334153, ... See A135720 > 561.

The smallest such semiprimes are 341, 2701, ?, 721801, ... Cf. A285549.

Examples

			The first Fermat pseudoprime to base 2 is 341, and 341 is not a Fermat pseudoprime to base 3, so a(1) = 3.
		

Crossrefs

Cf. A001567, A002997, A083876, A135720, A177415, A214305, A285549.

A141710 is a subsequence.

Programs

Mathematica

a[p_] := Module[{m=3}, While[Mod[m^(p-1), p] == 1, m++]; m]; psp = Select[Range[3, 1000000, 2], CompositeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &]; Map[a, psp] (* Amiram Eldar, Nov 19 2018 *)

Extensions

More terms from Amiram Eldar, Nov 19 2018