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 A141768 Odd numbers with increasing numbers of bases to which they are strong pseudoprimes. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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

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Last modified April 17 03:58 EDT 2021. Contains 343059 sequences. (Running on oeis4.)