Euler-Jacobi Pseudoprime
An Euler-Jacobi pseudoprime to a base 
is an Odd Composite numbers such that 
and the JacobiSymbol 
satisfies
(Guy 1994; but note that Guy calls these simply ``Euler pseudoprimes''). No Odd Composite number is an Euler-Jacobipseudoprime 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, ... (Sloane's A047713).See also Euler Pseudoprime, Pseudoprime
References
Guy, R. K. ``Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes.'' §A12 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994. Sloane, N. J. A. SequenceA047713/M5461in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
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