An Euler-Jacobi pseudoprime to a base is an odd composite number such that and the Jacobi symbol satisfies
(Guy 1994; but note that Guy calls these simply "Euler pseudoprimes"). No odd composite number is an Euler-Jacobi pseudoprime for all bases relatively prime to it. This class includes some Carmichael numbers, all strong pseudoprimes to base , and all Euler pseudoprimes to base . An Euler pseudoprime is pseudoprime to at most 1/2 of all possible bases less than itself.
The first few base-2 Euler-Jacobi pseudoprimes are 561, 1105, 1729, 1905, 2047, 2465, ... (OEIS A047713), and the first few base-3 Euler-Jacobi pseudoprimes are 121, 703, 1729, 1891, 2821, 3281, 7381, ... (OEIS A048950). The number of base-2 Euler-Jacobi primes less than , , ... are 0, 1, 12, 36, 114, ... (OEIS A055551).