A101036 -id:A101036 - cp's OEIS frontend

A306151 Let k be a Sierpiński or Riesel number, and let p be the largest number in a set of n primes which cover every number of the form k2^m + 1 (or of the form k2^m - 1) with m >= 1. a(n) = 0 if no covering set with n primes exists, otherwise a(n) = p if and only if there exists no number k that has a covering set with precisely n primes and with largest prime < p.

0, 0, 0, 0, 0, 241, 73, 241, 151, 241, 151, 151, 241, 257, 257, 257

Offset: 1

Arkadiusz Wesolowski, Jun 23 2018

			Examples of the covering sets:
- for n = 6, the set is {3, 5, 7, 13, 17, 241},
- for n = 7, the set is {3, 5, 7, 13, 19, 37, 73},
- for n = 8, the set is {3, 5, 7, 17, 19, 37, 73, 241},
- for n = 9, the set is {3, 5, 7, 11, 13, 31, 41, 61, 151},
- for n = 10, the set is {3, 5, 7, 11, 17, 31, 41, 61, 151, 241},
- for n = 11, the set is {3, 5, 7, 11, 19, 31, 37, 41, 61, 73, 151},
- for n = 12, the set is {3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 151},
- for n = 13, the set is {3, 7, 11, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241},
- for n = 14, the set is {3, 7, 11, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 257},
- for n = 15, the set is {3, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
- for n = 16, the set is {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257}.

Cf. A076336, A101036, A206001, A206430, A305473.

Corrected by Arkadiusz Wesolowski, Aug 04 2023

A361898 A set of 13 primes that form a covering set for a Sierpiński (or Riesel) number.

Original entry on oeis.org

3, 5, 7, 11, 31, 73, 97, 151, 241, 631, 673, 1321, 23311

Offset: 1

Arkadiusz Wesolowski, Mar 28 2023

A set of primes is called the covering set for the Sierpiński number k if for every positive integer m there is at least one prime in the set which divides k*2^m + 1. Similarly, a set of primes is called the covering set for the Riesel number j if for every positive integer m there is at least one prime in the set which divides j*2^m - 1.

Subset of A296924.

Cf. A076336, A101036, A291360, A361899, A361900.

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

Original entry on oeis.org

1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479

Offset: 1

Thomas Ordowski, Jan 22 2024

nonn

Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.
If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.
For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).

			7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.

Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.

fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)

For n > 1, a(n) = A138290(n-1) + 1.

A208083(a(n)) = 0, for n > 0.

More terms from Robert G. Wilson v, Jan 22 2024

A179289 Smallest index k such that prime(k)*2^n-1 is prime, or zero if there is no prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 2, 3, 4, 3, 2, 1, 11, 3, 22, 1, 4, 1, 18, 7, 4, 23, 6, 23, 18, 5, 44, 23, 4, 1, 14, 3, 11, 2, 11, 7, 11, 2, 18, 28, 8, 16, 2, 102, 4, 9, 11, 3, 8, 5, 174, 24, 63, 3, 2, 103, 22, 23, 130, 1, 22, 16, 18, 2, 19, 55, 14, 41, 34, 15

Offset: 1

Pierre CAMI, Jul 09 2010

nonn

Define partial sum S(N)=Sum(n=1...N) of n , T(N)=Sum(n=1...N) of k(n) The ratio T(N)/S(N) --> approx 0.5236 as N --> infinity.

It is conjectured that a(42228) is the first 0 term. This corresponds to the first Riesel number, 509203, which happens to be prime. See A101036. - T. D. Noe, Mar 23 2011

Cf. A126715.

sik[n_]:=Module[{c=2^n,k=1},While[!PrimeQ[Prime[k]*c-1],k++];k]; Array[ sik,70] (* The program will NOT identify cases where no prime satisfies the definition.  See the second comment. *) (* Harvey P. Dale, Jan 10 2016 *)

SCRIPT / DIM nn,0 / DIM kk / DIMS st / LABEL loopn / SET nn,nn+1 / IF nn>10000 THEN END / SET kk,0 / LABEL loopk / SET kk,kk+1 / SET st,%d,%d,%d\,;nn;kk;p(kk) / PRP p(kk)*2^nn-1 / IF ISPRIME THEN GOTO loopn / GOTO loopk / / This file is the in.txt file / The command is PFGW -f in.txt / The results are in the file pfgw-prime.log for small n / and in the pfgw.log file for greatest n / Program PFGW from Primeform Group /

a(n) = {my(k=1); while (!isprime(prime(k)*2^n-1), k++); k;} \\ Michel Marcus, Sep 16 2019

a(n) = 1 for n = A000043(k) - 1, the Mersenne exponents minus 1. - T. D. Noe, Mar 23 2011

A187467 Least k > 1 such that prime(k)*2^n - 1 is prime, or zero if never prime.

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 3, 4, 3, 2, 3, 11, 3, 22, 7, 4, 2, 18, 7, 4, 23, 6, 23, 18, 5, 44, 23, 4, 98, 14, 3, 11, 2, 11, 7, 11, 2, 18, 28, 8, 16, 2, 102, 4, 9, 11, 3, 8, 5, 174, 24, 63, 3, 2, 103, 22, 23, 130, 7, 22, 16, 18, 2

Offset: 1

Pierre CAMI, Mar 10 2011

nonn

As N increases, it appears that (Sum_{i=1..N} a(i)) / (Sum_{i=1..N} i) tends to 1/2, i.e., the partial sums grow roughly proportional to the triangular numbers.

It is conjectured that a(42228) is the first 0 term. This corresponds to the first Riesel number, 509203, which happens to be prime. See A101036. - T. D. Noe, Mar 23 2011

Pierre CAMI, Table of n, a(n) for n = 1..4100

Cf. A101036, A126715, A187468.

A187467 := proc(n) local k; for k from 2 do if isprime( ithprime(k)*2^n-1) then return k; end if; end do: end proc: # R. J. Mathar, Mar 19 2011

a(n) = primepi(A126715(n)). - T. D. Noe, Mar 10 2011

a(n) >= A179289(n). - R. J. Mathar, Mar 19 2011

A206430 Least k such that k*2^m - 1 has a covering set with precisely n primes.

Original entry on oeis.org

509203, 777149, 7106977, 60014203

Offset: 6

Arkadiusz Wesolowski, Apr 19 2012

A set of primes is a covering set for k*2^m - 1 if for every positive integer m there is some prime in the set which divides k*2^m - 1. Only minimal covering sets are considered here (those which would not remain covering sets with the removal of any element).

			509203 has the covering set {3, 5, 7, 13, 17, 241}.
777149 has the covering set {3, 5, 7, 13, 19, 37, 73}.
7106977 has the covering set {3, 5, 13, 17, 19, 109, 241, 433}.
60014203 has the covering set {3, 5, 7, 11, 13, 31, 41, 61, 151}.

Wikipedia, Covering set
Wikipedia, Riesel number

Cf. A076337, A101036.

A251757 Odd numbers n not congruent to 1 mod 6 and that are not perfect powers such that for all k >= 1 the numbers n*4^k - 1 are composite.

Original entry on oeis.org

39939, 656709, 762701, 777149, 779619, 790841, 869859, 1106681, 1156299, 1254341, 1320981, 1330319, 1730681, 2251349, 2344211, 2882541, 2924861, 3079469, 3292241, 3419789, 3580901, 3661529, 3781541, 3784439, 3848289, 4358481, 4384979, 4507889, 4524549

Offset: 1

Arkadiusz Wesolowski, Dec 15 2014

nonn

a(n) is a Riesel number if it is of the form 6*m + 5.

Cf. A101036, A251057.

A269326 Let k be a number which is simultaneously Sierpiński and Riesel, and let P be a set of primes which cover every number of the form k2^m + 1 and of the form k2^m - 1 with m >= 1. Sequence shows elements of the set P which has the property that the product of its primes is as small as it is possible.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331

Offset: 1

Arkadiusz Wesolowski, Feb 23 2016

Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.

Cf. A076335, A076336, A101036.

PrimeDivisors((2^36-1)*(2^48-1)*(2^60-1))[1..18];

A271027 a(n) = 3661529 + 11184810*n.

Original entry on oeis.org

3661529, 14846339, 26031149, 37215959, 48400769, 59585579, 70770389, 81955199, 93140009, 104324819, 115509629, 126694439, 137879249, 149064059, 160248869, 171433679, 182618489, 193803299, 204988109, 216172919, 227357729, 238542539, 249727349, 260912159, 272096969, 283281779, 294466589

Offset: 0

Altug Alkan, Mar 29 2016

a(n) and a(n) + 14 are the members of A101036.

14 appears as a minimum difference between Riesel numbers for the first 15000 terms that are listed in b-file of A101036.

			a(1) = 3661529 + 11184810*1 = 14846339.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A101036, A244070, A270993.

[3661529 + 11184810*n : n in [0..40]]; // Wesley Ivan Hurt, Apr 02 2016

A271027:=n->3661529 + 11184810*n: seq(A271027(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016

CoefficientList[Series[(3661529 + 7523281 x)/(1 - x)^2, {x, 0, 26}], x] (* Michael De Vlieger, Mar 29 2016 *)
LinearRecurrence[{2,-1},{3661529,14846339},30] (* Harvey P. Dale, Sep 10 2019 *)

PARI
```
a(n) = 3661529 + 11184810*n;
```

x='x+O('x^99); Vec((3661529+7523281*x)/(1-x)^2)

for n in range(0,100):print(3661529+11184810*n) # Soumil Mandal, Apr 03 2016

G.f.: (3661529 + 7523281*x)/(1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>1.

A283806 Odd numbers which are uniquely decomposable into the sum of a prime and a power of two.

Original entry on oeis.org

3, 5, 17, 29, 41, 53, 59, 65, 89, 97, 119, 137, 163, 179, 185, 191, 193, 209, 217, 219, 221, 223, 233, 239, 247, 253, 269, 281, 305, 307, 311, 343, 359, 389, 403, 407, 415, 419, 427, 431, 457, 491, 505, 521, 533, 545, 547, 557, 569, 575, 581, 583, 597, 613, 637

Offset: 1

Arkadiusz Wesolowski, Mar 17 2017

It is conjectured that none of these numbers is in A101036.
A positive integer n belongs to this sequence if n is of the form x*y + x - 1 and for some m >= 1:
1) y = -1 + 2 * Product_{k=0..m} (2^(2^k) + 1),
2) x <= 2^(2^(m+1) - 1),
3) n - 2^(2^(m+1)) is prime.
Odd numbers m that satisfy A109925(m) = 1. - Michel Marcus, Mar 19 2017

			17 is in the sequence since 17 - 2^2 = 13 is a prime and 17 - 2^0 = 16, 17 - 2^1 = 15, 17 - 2^3 = 9, 17 - 2^4 = 1 are all nonprimes.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Wikipedia, Prime number

Cf. A000215, A109925.

lst:=[]; for n in [1..637 by 2] do c:=0; r:=Floor(Log(n)/Log(2)); for x in [0..r] do a:=n-2^x; if IsPrime(a) then c+:=1; end if; if c eq 2 then break; end if; end for; if c eq 1 then Append(~lst, n); end if; end for; lst;

Select[Range[1, 640, 2], Function[n, Total@ Boole@ PrimeQ@ Map[n - # &, 2^Range[0, Floor@ Log2@ n]] == 1]] (* Michael De Vlieger, Mar 18 2017 *)

isok(n) = (n % 2) && (sum(k=0, log(n)\log(2), isprime(n-2^k)) == 1); \\ Michel Marcus, Mar 18 2017

from sympy import isprime
import math
print([n for n in range(1001) if n%2 and sum([isprime(n-2**k) for k in range(int(math.floor(math.log(n)/math.log(2))) + 1)]) == 1]) # Indranil Ghosh, Mar 29 2017

a(n) ~ 10*(n + n/log(n)).

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A361898 A set of 13 primes that form a covering set for a Sierpiński (or Riesel) number.

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A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

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A179289 Smallest index k such that prime(k)*2^n-1 is prime, or zero if there is no prime.

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A187467 Least k > 1 such that prime(k)*2^n - 1 is prime, or zero if never prime.

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A206430 Least k such that k*2^m - 1 has a covering set with precisely n primes.

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A251757 Odd numbers n not congruent to 1 mod 6 and that are not perfect powers such that for all k >= 1 the numbers n*4^k - 1 are composite.

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A271027 a(n) = 3661529 + 11184810*n.

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