

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


LINKS



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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



