The Euler numbers, also called the Secant Numbers or Zig Numbers, are
defined for by

(1) |

(2) |

Some values of the Euler numbers are

(Sloane's A000364). The first few Prime Euler numbers occur for , 3, 19, 227, 255, ... (Sloane's A014547) up to a search limit of .

The slightly different convention defined by

(3) | |||

(4) |

is frequently used. These are, for example, the Euler numbers computed by the

(5) |

(6) |

To confuse matters further, the Euler Characteristic is sometimes also called the ``Euler number.''

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.''
§23.1 in *Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing.* New York: Dover, pp. 804-806, 1972.

Conway, J. H. and Guy, R. K. In *The Book of Numbers.* New York: Springer-Verlag, pp. 110-111, 1996.

Guy, R. K. ``Euler Numbers.'' §B45 in *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, p. 101, 1994.

Knuth, D. E. and Buckholtz, T. J. ``Computation of Tangent, Euler, and Bernoulli Numbers.'' *Math. Comput.* **21**, 663-688, 1967.

Sloane, N. J. A. Sequences
A014547 and
A000364/M4019
in ``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.

Spanier, J. and Oldham, K. B. ``The Euler Numbers, .'' Ch. 5 in *An Atlas of Functions.*
Washington, DC: Hemisphere, pp. 39-42, 1987.

© 1996-9

1999-05-25