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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A187716 Odd numbers m divisible by 3 such that for every k >= 1, m*2^k + 1 has a divisor in the set {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331}.

Original entry on oeis.org

21484572547591559649, 50166404682516122859, 51814002736113272553, 53246606581410442023, 58992081042572747991, 65634687179877002283, 80269357428943941837, 92027572854849003627, 103083799330841020677
Offset: 1

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Author

Arkadiusz Wesolowski, Mar 17 2011

Keywords

Comments

Wilfrid Keller (2004, published) gave the first known example.
21484572547591559649 computed in 2017 by the author.
Conjecture: 21484572547591559649 is the smallest Sierpiński number that is divisible by 3. - Arkadiusz Wesolowski, May 12 2017
The above conjecture is false, because the Sierpiński number 7592506760633776533 is a counterexample. - Arkadiusz Wesolowski, Jul 27 2023

Links

Crossrefs

Cf. A076336, A187714.

Extensions

Name changed and entry revised by Arkadiusz Wesolowski, May 11 2017

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

Original entry on oeis.org

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

Views

Author

Arkadiusz Wesolowski, Jun 13 2012

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A076337, A101036, A187714, A222534.

Extensions

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

A305473 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.

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

Views

Author

Arkadiusz Wesolowski, Jun 02 2018

Keywords

Comments

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

			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}.

References

  • 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.

Links

Crossrefs

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

Formula

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.

A237592 Odd numbers n divisible by 3 such that for all k >= 1 the numbers n*2^k - 1 and n*2^k + 1 do not form a twin prime pair.

Original entry on oeis.org

237, 807, 4581, 32469, 41091, 60981, 62637, 63351, 76593, 80979, 84387, 85047, 92343, 93621, 96891, 102183, 113679, 123609, 130629, 139647, 140571, 158883, 171837, 172857, 177201, 178401, 184251, 205347, 207243, 212547, 217011, 219291, 220851, 238779, 250401
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 22 2014

Keywords

Links

Crossrefs

Supersequence of A187714 and of A187716.
Showing 1-4 of 4 results.

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