A101036 -id:A101036 - cp's OEIS frontend

A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

144323411864333, 175321252530209, 190779128601685, 316031956469111, 389882208980861, 450590081221877, 2420018284798363, 2715458757443051, 3161282469971861, 3366332338600025, 3643757921262355, 4380746955320089, 4409682697067321, 5089175909950511, 5281690092088615

Offset: 1

Arkadiusz Wesolowski, Jul 23 2023

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34560

Cf. A101036, A076335, A180247, A291360, A364413.

For n > 34560, a(n) = a(n-34560) + 10014447295554878022.

A236321 Conjectured number of Riesel numbers less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 65, 668, 6711

Offset: 1

Arkadiusz Wesolowski, Jan 22 2014

Wikipedia, Riesel number

Cf. A076337, A101036, A236320, A237880.

a(n) ~ A236320(n).

A237880 Conjectured number of distinct integers < 10^n that are Sierpiński or Riesel or simultaneously Sierpiński and Riesel numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 16, 134, 1345, 13420

Offset: 1

Arkadiusz Wesolowski, Feb 14 2014

Wikipedia, Riesel number
Wikipedia, Sierpinski number

Cf. A076335, A076336, A076337, A101036, A236320, A236321.

a(n) = A236320(n) + A236321(n) for n <= 9.

Definition clarified by Arkadiusz Wesolowski, Jun 05 2021

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Eric Chen, Dec 14 2014

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.
a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.
a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.

Jinyuan Wang, Table of n, a(n) for n = 1..1146

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

a(19) corrected by Jinyuan Wang, Mar 25 2023

A258154 Least k such that k2^m - 1 has a covering set of modulus 2n, or 0 if no such value exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 509203, 0, 0, 0, 0, 0, 777149, 0, 0, 0, 0, 0, 10157893, 0, 0, 0, 0, 0, 60014203, 0, 200883553191612793, 0, 0, 0, 7106977, 0, 0, 0

Offset: 1

Arkadiusz Wesolowski, Nov 05 2015

nonn

Wikipedia, Riesel number

Cf. A101036, A257647, A257861.

A263392 Riesel numbers that have at least two covering sets.

Original entry on oeis.org

66837283, 68026001, 71344361, 71604733, 74477159, 94270439, 97579567, 122606647, 140022313, 153196541, 170969579, 182720077, 198885523, 211327993, 220316413, 227116031, 280536071, 285155137, 290419777, 292202437, 293175191, 309240779, 324576431, 331006603

Offset: 1

Arkadiusz Wesolowski, Oct 16 2015

nonn

			For every k >= 1, 97579567*2^k - 1 has a divisor in the set {3, 5, 7, 13, 17, 241} and also in the set {3, 5, 13, 17, 97, 241, 257}. 97579567 is therefore in the sequence.

Wikipedia, Riesel number

Cf. A101036, A263391.

1 term inserted by Arkadiusz Wesolowski, Aug 28 2016

More terms from Arkadiusz Wesolowski, Jan 09 2018

A263562 Primes p such that for every k >= 1, p*2^k - 1 has a divisor in the set {3, 5, 13, 17, 97, 241, 257}.

Original entry on oeis.org

1865978047, 1889699677, 2362339121, 3637126963, 11776639499, 19321614419, 20000692169, 20111311169, 20592473107, 20597584901, 21477425107, 23368396573, 23479945327, 25326720611, 26161244323, 27190405961, 27380064223, 27474950743, 31467088979

Offset: 1

Arkadiusz Wesolowski, Oct 21 2015

nonn

What is the smallest term of this sequence that belongs to A180247? Is it the smallest prime Brier number?

Chris Caldwell, The Prime Glossary, Riesel number
Carlos Rivera, Problem 52

Cf. A101036, A180247, A263560.

Subsequence of A263561.

A273987 Smallest Riesel number to base n.

Original entry on oeis.org

509203, 63064644938, 9, 346802, 84687, 408034255082, 14, 4, 10176, 862, 25, 302, 4, 36370321851498, 9, 86, 246, 144, 8, 560, 4461, 476, 4, 36, 149, 8, 144, 4, 1369, 134718, 10, 16, 6, 287860, 4, 7772, 13, 4, 81, 8, 15137, 672, 4, 22564, 8177, 14, 3226, 36, 16

Offset: 2

Tim Johannes Ohrtmann, Jun 06 2016

a(2), a(3), a(5), a(6), a(7), a(10), a(15), a(22), a(23), a(30), ... are only conjectural (see links).

Gary Barnes, Riesel conjectures and proofs.
Gary Barnes, Riesel conjectures and proofs, Powers of 2.
Gary Barnes, Condensed table of all conjectures for bases ≤ 1030.
Wikipedia, Riesel number.

Cf. A076337, A101036.

A276417 a(n) = least positive k such that (2*n + 1) - 2^k is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 0, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 0, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 0

Arkadiusz Wesolowski, Sep 02 2016

nonn

a(n) = 1 iff n is in A006254. - Robert Israel, Sep 02 2016

For n > 1, a(n) = 0 iff 2n+1 is de Polignac number, A006285. - Thomas Ordowski, Apr 13 2017

			a(14) = 4 because (2*14 + 1) - 2^k is composite for k = 1, 2, 3 and prime for k = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A006254, A006285, A101036, A133122, A188903, A232460, A252168.

lst:=[]; for n in [1..173 by 2] do k:=0; c:=k; repeat k+:=1; c+:=1; a:=n-2^k; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, 0); else Append(~lst, c); end if; end for; lst;

f:= proc(n) local k;
     for k from 1 do
       if 2*n < 2^k then return 0
       elif isprime(2*n+1-2^k) then return k
       fi
     od
end proc:
map(f, [$0..100]); # Robert Israel, Sep 02 2016

Table[If[n <= 2, 0, k = 1; While[! PrimeQ[2 n + 1 - 2^k], k++]; k], {n, 0, 120}] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = my(k=1); while(2^k < 2*n+1, if(ispseudoprime((2*n+1)-2^k), return(k)); k++); return(0) \\ Felix Fröhlich, Sep 02 2016

If A188903(n) >= 2, then a(n) = log_2(A188903(n)), otherwise a(n) = 0.

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

A364412 Odd numbers m such that for every k >= 1, m*2^k - 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

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A236321 Conjectured number of Riesel numbers less than 10^n.

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A237880 Conjectured number of distinct integers < 10^n that are Sierpiński or Riesel or simultaneously Sierpiński and Riesel numbers.

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A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

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A258154 Least k such that k*2^m - 1 has a covering set of modulus 2*n, or 0 if no such value exists.

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A263392 Riesel numbers that have at least two covering sets.

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A263562 Primes p such that for every k >= 1, p*2^k - 1 has a divisor in the set {3, 5, 13, 17, 97, 241, 257}.

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A273987 Smallest Riesel number to base n.

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A276417 a(n) = least positive k such that (2*n + 1) - 2^k is prime, or 0 if no such k exists.

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A258154 Least k such that k2^m - 1 has a covering set of modulus 2n, or 0 if no such value exists.