

A090659


Odd composites with increasing proportion of nontrivial nonwitnesses of compositeness by the MillerRabin primality test.


1



25, 91, 703, 1891, 12403, 38503, 79003, 88831, 146611, 188191, 218791, 269011, 286903, 385003, 497503, 597871, 736291, 765703, 954271, 1056331, 1314631, 1869211, 2741311, 3270403, 3913003, 4255903, 4686391, 5292631, 6186403, 6969511, 8086231, 9080191
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OFFSET

1,1


COMMENTS

Rabin has shown that the proportion has an upper bound of 0.25. If the trivial nonwitnesses are counted, this upper bound is reached at 9. If the conjecture is true that the later terms are always the product of two primes p and (2*p1), then the sequence continues 188191 218791 269011 286903 385003 497503 597871 736291 765703 954271 1056331 1314631 1869211 2741311 3270403 3913003 4255903 4686391 5292631.
Dickson's conjecture implies that this sequence is infinite. Can this be proved unconditionally?  Charles R Greathouse IV, Mar 10 2011
Higgins' conjecture 2 is implied by his conjecture 1, which is true by the general form of the number of nonwitnesses of an odd number.  Charles R Greathouse IV, Mar 10 2011


REFERENCES

S. Narayanan, Improving the Speed and Accuracy of the MillerRabin Primality Test 2015; http://math.mit.edu/research/highschool/primes/materials/2014/Narayanan.pdf


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..5411
Brian C. Higgins, The RabinMiller Primality Test: Some Results on the Number of Nonwitnesses to Compositeness
Michael O. Rabin, Probabilistic algorithm for testing primality, Journal of Number Theory 12:1 (1980), pp. 128138.


EXAMPLE

25 has 2 nontrivial nonwitnesses (NTNW), namely (7,18), for a proportion of 2/22=0.0909. The denominator is 22 because the nonwitnesses are selected from 2..23 (as 1 and 24 are trivial nonwitnesses).
49 has 4 NTNW, namely (18,19,30,31) for a proportion of 4/46=0.0870. This is a smaller proportion than 0.0909 for 25.
91=7*13 has 16 NTNW in the range [2..89], namely [9, 10, 12, 16, 17, 22, 29, 38, 53, 62, 69, 74, 75, 79, 81, 82], for a proportion of 16/88=0.182. It also has two trivial nonwitnesses 1 and 90, which are not counted. The next integer with a higher proportion is 703, with 160 nontrivial nonwitnesses and proportion 0.229.


CROSSREFS

Subsequence of A141768.
Sequence in context: A044212 A044593 A262492 * A010013 A256716 A063769
Adjacent sequences: A090656 A090657 A090658 * A090660 A090661 A090662


KEYWORD

nonn,changed


AUTHOR

Ken Takusagawa, Dec 14 2003


EXTENSIONS

Extended and edited by Charles R Greathouse IV, Mar 09 2011


STATUS

approved



