

A123159


Conjectured smallest Sierpiński numbers of the second kind S, base b=2,3,4,5,..., where S*b^n+1 is composite for all n>=1 and gcd(S+1, b1) = 1.


10



78557, 125050976086, 66741, 159986, 174308, 1112646039348, 1, 2344, 9175, 1490, 521, 132, 4, 91218919470156, 2500, 278, 398, 765174, 8, 1002, 6694, 182, 30651, 262638, 221, 8, 4554, 4, 867, 6360528, 1, 1854, 6, 214018, 1886, 2604, 14, 166134, 826477, 8, 13372, 2256, 4, 53474
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OFFSET

2,1


COMMENTS

Some values with base b=2^x+1 for integers x have also been calculated  see the links.


LINKS

Table of n, a(n) for n=2..45.
G. Jaeschke, On the smallest k such that all k * 2^n +1 are composite, Math. Comp., 40:181 (1983) 381384. MR 84k:10006.
Primeform Group, base=2^x+1.
Primeform Group, base=3.
Primeform Group, base=5.
Carlos Rivera, Problem 36. The LiskovetsGallot numbers, The Prime Puzzles and Problems Connection.
Guido Smetrijns, Andrew J. Walker, Shane Findley, Jim Fougeron, Mikael Klasson, Robert Smith and others, Riesel/Sierpinski in base 3, digest of 43 messages in primeform Yahoo group, May 24, 2004  Jan 7, 2007. [Cached copy]
Robert Smith, David Broadhurst, Shane Findley, Sierpinski / Riesel base 2^x+1, digest of 4 messages in primeform Yahoo group, Sep 26  Sep 27, 2004. [Cached copy]
Robert Smith, Guido Smetrijns, Mikael Klasson, Riesel Sierpinski in base 5, digest of 6 messages in primeform Yahoo group, Sep 17  Sep 18, 2004


EXAMPLE

For base=3, S+1 should be coprime to 31 = 2, so S must be even. Find a covering set of multiplicative orders of primes base b and discover S by trial and error using the Chinese Remainder Theorem.


CROSSREFS

Cf. A076336.
Sequence in context: A038815 A076336 A244562 * A184230 A186612 A180973
Adjacent sequences: A123156 A123157 A123158 * A123160 A123161 A123162


KEYWORD

nonn


AUTHOR

Robert Smith (robert_smith44(AT)hotmail.com), Oct 02 2006


EXTENSIONS

a(6) from Arkadiusz Wesolowski, Nov 20 2014
a(3) corrected and a(7)a(45) from Eric Chen, Dec 16 2014


STATUS

approved



