|
%I #59 Nov 14 2022 23:04:43
%S 73,257,151,151,257,73,151,1321,73,109,1321,73,151,257,73,73,331,257,
%T 109,331,73,73,1321,73,151,331,73,241
%N Let k be a Sierpiński or Riesel number divisible by 2*n - 1, and let p be the largest number in a set of primes which cover every number of the form k*2^m + 1 (or of the form k*2^m - 1) with m >= 1. a(n) = p if and only if there exists no number k that has a covering set with largest prime < p.
%C R. G. Stanton found that a(2) = 257.
%C a(n) >= 73 for any n, see [Stanton].
%C There exist infinitely many Riesel numbers that are divisible by 15. The number 334437671621489828385689959795356586832846847109919809460835 is one such number.
%D R. G. Stanton, Further results on covering integers of the form 1 + k * 2^n by primes, pp. 107-114 in: Kevin L. McAvaney (ed.), Combinatorial Mathematics VIII, Lecture Notes in Mathematics 884, Berlin: Springer, 1981.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_049.htm">Problem 49. Sierpinski-like numbers</a>, The Prime Puzzles and Problems Connection.
%F a(((2*n-1)^b+1)/2) = a(n) for every b >= 2.
%F a((2*b-1)*n-b+1) >= a(n) for every b >= 2; n > 1.
%F a(n) = 73 if and only if gcd(2*n-1, 70050435) = 1.
%e Examples of the covering sets:
%e - for n = 2, the set is {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
%e - for n = 3, the set is {3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 151},
%e - for n = 4, the set is {3, 5, 11, 13, 19, 31, 37, 41, 61, 73, 151},
%e - for n = 6, the set is {3, 5, 7, 13, 19, 37, 73},
%e - for n = 7, the set is {3, 5, 7, 11, 19, 31, 37, 41, 61, 73, 151},
%e - for n = 8, the set is {7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 71, 73, 97, 109, 113, 127, 151, 193, 211, 241, 257, 281, 331, 337, 421, 433, 577, 673, 1153, 1321},
%e - for n = 11, the set is {5, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 181, 193, 241, 257, 331, 433, 577, 631, 673, 1153, 1321},
%e - for n = 17, the set is {5, 7, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
%e - for n = 18, the set is {3, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
%e - for n = 20, the set is {5, 7, 11, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
%e - for n = 26, the set is {5, 7, 11, 13, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
%e - for n = 28, the set is {3, 7, 13, 17, 19, 37, 73, 109, 241}.
%Y Cf. A076336, A101036, A187714, A187716, A213529, A222534, A244071, A244562, A306151.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Jun 02 2018
|
