A222534 -id:A222534 - cp's OEIS frontend

A213529 Smallest Riesel number that is divisible by the n-th prime.

7148695169714208807, 84392786545, 42270067, 1254341, 514389187, 16861093, 1730653, 1730681, 4485343, 790841, 15692699, 992077, 2136283, 1730681, 24683107, 9666029, 9560713, 33282853, 9375479, 14604599, 1247173, 19437853, 34546507, 790841, 3781541, 1715053, 17710319, 45501941

Offset: 2

Arkadiusz Wesolowski, Jun 13 2012

nonn

Some examples of Riesel numbers that are divisible by 3 are in A187714.

For an odd prime p and odd k, if p divides k, then p does not divide k*2^n - 1 for any n.

			1254341 is first Riesel number that is divisible by 11, the 5th prime - so a(5) = 1254341.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary, Riesel number

Cf. A076337, A101036, A187714, A222534.

Offset changed and initial term added by Arkadiusz Wesolowski, May 11 2017

A305473 Let k be a Sierpiński or Riesel number divisible by 2n - 1, and let p be the largest number in a set of primes which cover every number of the form k2^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.

Original entry on oeis.org

73, 257, 151, 151, 257, 73, 151, 1321, 73, 109, 1321, 73, 151, 257, 73, 73, 331, 257, 109, 331, 73, 73, 1321, 73, 151, 331, 73, 241

Offset: 1

Arkadiusz Wesolowski, Jun 02 2018

nonn

R. G. Stanton found that a(2) = 257.
a(n) >= 73 for any n, see [Stanton].
There exist infinitely many Riesel numbers that are divisible by 15. The number 334437671621489828385689959795356586832846847109919809460835 is one such number.

			Examples of the covering sets:
- for n = 2, the set is {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
- for n = 3, the set is {3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 151},
- for n = 4, the set is {3, 5, 11, 13, 19, 31, 37, 41, 61, 73, 151},
- for n = 6, the set is {3, 5, 7, 13, 19, 37, 73},
- for n = 7, the set is {3, 5, 7, 11, 19, 31, 37, 41, 61, 73, 151},
- 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},
- 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},
- for n = 17, the set is {5, 7, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 18, the set is {3, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
- for n = 20, the set is {5, 7, 11, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 26, the set is {5, 7, 11, 13, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 28, the set is {3, 7, 13, 17, 19, 37, 73, 109, 241}.

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.

Carlos Rivera, Problem 49. Sierpinski-like numbers, The Prime Puzzles and Problems Connection.

Cf. A076336, A101036, A187714, A187716, A213529, A222534, A244071, A244562, A306151.

a(((2*n-1)^b+1)/2) = a(n) for every b >= 2.
a((2*b-1)*n-b+1) >= a(n) for every b >= 2; n > 1.
a(n) = 73 if and only if gcd(2*n-1, 70050435) = 1.

cp's OEIS Frontend

A213529 Smallest Riesel number that is divisible by the n-th prime.

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