A187716 -id:A187716 - cp's OEIS frontend

A187714 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}.

7148695169714208807, 17968583418362170239, 26363076126393718191, 57376760867272385247, 67950587841687767283, 73873959473901564111, 81055172741266754727, 96217896533288105991, 104173338506128098489

Offset: 1

Arkadiusz Wesolowski, Mar 17 2011

nonn

Wilfrid Keller (2004, published) gave the first known example.
7148695169714208807 computed in 2017 by the author.
Conjecture: 7148695169714208807 is the smallest Riesel number that is divisible by 3. - Arkadiusz Wesolowski, May 12 2017

Chris Caldwell, The Prime Glossary, Riesel number
Carlos Rivera, Problem 49

Cf. A101036, A187716.

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

A222534 Smallest Sierpinski number that is divisible by the n-th prime.

Original entry on oeis.org

7592506760633776533, 36293948155, 157957457, 603713, 422590909, 78557, 6134663, 1259779, 575041, 7892569, 2931991, 4095859, 2541601, 7892569, 29169451, 271577, 35193889, 12824269, 603713, 9454157, 575041, 7696009, 5455789, 41561687, 7400371, 2191531, 29046541, 2931991

Offset: 2

Arkadiusz Wesolowski, Feb 24 2013

nonn

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

			603713 is first Sierpinski number that is divisible by 11, the 5th prime - so a(5) = 603713.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary: Sierpinski number
Carlos Rivera, Problem 86. A smaller Sierpinski integer divided by 3, The Prime Puzzles & Problems Connection.

Cf. A076336, A187716, A213529, A364413.

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

a(2) corrected by Arkadiusz Wesolowski, Jul 27 2023

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.

A237592 Odd numbers n divisible by 3 such that for all k >= 1 the numbers n2^k - 1 and n2^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

Arkadiusz Wesolowski, Apr 22 2014

nonn

Eric Weisstein's World of Mathematics, Twin Primes

Supersequence of A187714 and of A187716.

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A187714 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}.

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A222534 Smallest Sierpinski number that is divisible by the n-th prime.

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A237592 Odd numbers n divisible by 3 such that for all k >= 1 the numbers n2^k - 1 and n2^k + 1 do not form a twin prime pair.

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cp's OEIS Frontend

A187714 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}.

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A222534 Smallest Sierpinski number that is divisible by the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

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.

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A237592 Odd numbers n divisible by 3 such that for all k >= 1 the numbers n2^k - 1 and n2^k + 1 do not form a twin prime pair.