A141768 Odd numbers with increasing numbers of bases to which they are strong pseudoprimes.

9, 25, 49, 91, 341, 481, 703, 1541, 1891, 2701, 5461, 6533, 8911, 12403, 18721, 29341, 31621, 38503, 79003, 88831, 146611, 188191, 218791, 269011, 286903, 385003, 497503, 597871, 736291, 765703, 954271, 1024651, 1056331, 1152271, 1314631

Offset

1,1

Comments

These numbers are the worst cases for the Rabin-Miller probable-prime test.

Alford, Granville, & Pomerance show that this sequence is infinite.

The sequence is unchanged whether one, both, or neither of 1 and n-1 are included as bases.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..5476

W. R. Alford, A. Granville, and C. Pomerance (1994). "On the difficulty of finding reliable witnesses". Lecture Notes in Computer Science 877, 1994, pp. 1-16.

Shyam Narayanan, Improving the Speed and Accuracy of the Miller-Rabin Primality Test

Shyam Narayanan, Improving the Accuracy of Primality Tests by Enhancing the Miller-Rabin Theorem (2014)

Michael O. Rabin, Probabilistic algorithm for testing primality, Journal of Number Theory 12:1 (1980), pp. 128-138.

Index entries for sequences related to pseudoprimes

Example

25 is a 1-, 7-, 18- and 24-strong pseudoprime and no odd number less than 25 has four or more bases to which it is a strong pseudoprime.

Prog

(PARI) star(n)={n--; n>>valuation(n, 2)};
bases(n)=my(f=factor(n)[, 1], nu=valuation(f[1]-1, 2), nn = star(n)); for(i=2, #f, nu = min(nu, valuation(f[i] - 1, 2)); ); (1 + (2^(#f * nu) - 1) / (2^#f - 1)) * prod(i=1, #f, gcd(nn, star(f[i])));
r=0; forstep(n=9, 1e8, 2, if(isprime(n), next); t=bases(n); if(t>r, r=t; print1(n", ")))

Crossrefs

Cf. A014233, A071294, A194946.

Sequence in context: A192775 A318737 A246331 * A339126 A176970 A110284

Adjacent sequences: A141765 A141766 A141767 * A141769 A141770 A141771

Keyword

nonn

Author

Charles R Greathouse IV Sep 15 2008

Extensions

Edited by Charles R Greathouse IV, Jul 23 2010

Status

approved