%I #81 Nov 16 2019 12:34:39
%S 78557,125050976086,66741,159986,174308,1112646039348,1,2344,9175,
%T 1490,521,132,4,91218919470156,2500,278,398,765174,8,1002,6694,182,
%U 30651,262638,221,8,4554,4,867,6360528,1,1854,6,214018,1886,2604,14,166134,826477,8,13372,2256,4,53474
%N 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, b-1) = 1.
%C Some values with base b=2^x+1 for integers x have also been calculated - see the links.
%H G. Jaeschke, <a href="http://dx.doi.org/10.1090/S0025-5718-1983-0679453-8">On the smallest k such that all k * 2^n +1 are composite</a>, Math. Comp., 40:181 (1983) 381-384. MR 84k:10006.
%H Primeform Group, <a href="http://groups.yahoo.com/group/primeform/message/4773">base=2^x+1</a>.
%H Primeform Group, <a href="http://groups.yahoo.com/group/primeform/message/7747">base=3</a>.
%H Primeform Group, <a href="http://groups.yahoo.com/group/primeform/message/4742">base=5</a>.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_036.htm">Problem 36. The Liskovets-Gallot numbers</a>, The Prime Puzzles and Problems Connection.
%H Guido Smetrijns, Andrew J. Walker, Shane Findley, Jim Fougeron, Mikael Klasson, Robert Smith and others, <a href="/A123159/a123159_1.txt">Riesel/Sierpinski in base 3</a>, digest of 43 messages in primeform Yahoo group, May 24, 2004 - Jan 7, 2007. [Cached copy]
%H Robert Smith, David Broadhurst, Shane Findley, <a href="/A123159/a123159_2.txt">Sierpinski / Riesel base 2^x+1</a>, digest of 4 messages in primeform Yahoo group, Sep 26 - Sep 27, 2004. [Cached copy]
%H Robert Smith, Guido Smetrijns, Mikael Klasson, <a href="/A123159/a123159.txt">Riesel Sierpinski in base 5</a>, digest of 6 messages in primeform Yahoo group, Sep 17 - Sep 18, 2004
%e For base=3, S+1 should be coprime to 3-1 = 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.
%Y Cf. A076336.
%K nonn
%O 2,1
%A Robert Smith (robert_smith44(AT)hotmail.com), Oct 02 2006
%E a(6) from _Arkadiusz Wesolowski_, Nov 20 2014
%E a(3) corrected and a(7)-a(45) from _Eric Chen_, Dec 16 2014