A076336 -id:A076336 - cp's OEIS frontend

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

A244565 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 37, 73, 109 }.

Original entry on oeis.org

322523, 12413281, 16921847, 27862127, 29095681, 35430841, 43925747, 47635073, 50273851, 56517767, 57816799, 59929127, 60666107, 63662611, 66887071, 69265069, 77564731, 83460571, 87376127, 104697533

Offset: 1

Pierre CAMI, Jun 30 2014

nonn

For n > 144, a(n) = a(n-144) + 803736570, the first 144 values are in the table.

Pierre CAMI, Table of n, a(n) for n = 1..144

Cf. A076336, A244074, A244561, A244562, A244563, A244564.

For n > 144, a(n) = a(n-144) + 803736570.

A251057 Odd numbers n not congruent to 5 mod 6 such that for all k >= 1 the numbers n*4^k + 1 are composite.

Original entry on oeis.org

66741, 271129, 308481, 327739, 436029, 482719, 575041, 636921, 934909, 965431, 1259541, 1259779, 1384059, 1518781, 1639459, 1997589, 2038371, 2131099, 2191531, 2397951, 2473929, 2541601, 2576089, 2931991, 2965569, 3098059, 3608251, 3885579, 3999399, 4095859

Offset: 1

Arkadiusz Wesolowski, Dec 15 2014

nonn

A123159(4) = a(1).

a(n) is a Sierpiński number if it is of the form 6*m + 1.

Cf. A076336, A123159, A251757.

A260349 a(n) = min(k : A046067((k+1)/2) = n).

Original entry on oeis.org

1, 3, 7, 17, 55, 59, 19, 167, 31, 311, 289, 227, 351, 203, 379, 197, 103, 1253, 829, 335, 211, 353, 649, 437, 1921, 1853, 2869, 917, 361, 263, 283, 1637, 1213, 3353, 1519, 797, 241, 1691, 259, 1391, 2503, 1109, 3859, 1857

Offset: 0

Hugo van der Sanden, Jul 23 2015

nonn

a(n) is the first odd number k for which k * 2^i + 1 is prime when i = n but composite for all i: 0 <= i < n, or 0 if no such k exists. Thus it is the first k for which A046067((k+1)/2) = n, and therefore also the first k for which you need to test the primality of exactly n values to show that it is not a Sierpiński number.

Jaeschke shows that for each n>0, the set {k : A046067((k+1)/2) = n} is infinite. - Jeppe Stig Nielsen, Jul 06 2020

			7 * 2^i + 1 is composite for i < 2 (with values 8, 15) but prime for i = 2 (29); the smaller odd numbers 1, 3 and 5 each yield a prime for smaller i, so a(2) = 7.

Hugo van der Sanden, Table of n, a(n) for n = 0..3253
G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.

Cf. A046067, A076336, A260350.

a(n)=forstep(k=1,+oo,2,for(i=0,n-1,ispseudoprime(k<Jeppe Stig Nielsen, Jul 06 2020

A263560 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

37158601, 7425967459, 9013226179, 13671059747, 14140683563, 17190420571, 17210867747, 18553286303, 18563509891, 19720992901, 20064786439, 22400387281, 23728062893, 29428753891, 36195177107, 41074421693, 44786947187, 45199948253, 48845530249

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, Sierpinski number
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.
Carlos Rivera, Problem 52

Cf. A076336, A180247, A263562.

Subsequence of A263347.

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.

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.

Original entry on oeis.org

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

A336347 Least prime factor of 44745755^4*2^(4n+2) + 1.

Original entry on oeis.org

13, 101, 29, 13, 39877, 41, 13, 37, 18661, 13, 41, 73, 13, 5719237, 144341, 13, 29, 89, 13, 353, 41, 13, 64450569241, 29, 13, 37, 101, 13, 89, 53, 13, 113, 313, 13, 37, 41, 13, 29, 73, 13, 41, 181, 13, 37, 29, 13, 857, 73, 13, 389, 41, 13, 37

Offset: 0

Jeppe Stig Nielsen, Jul 19 2020

nonn

There are k such that k*2^m + 1 is not prime for any m (then k is called a Sierpiński number). Erdős once conjectured that for such a k, the smallest prime factor of k*2^m + 1 would be bounded as m tends to infinitiy. The proven Sierpiński number k=44745755^4 is thought to be the first counterexample to this conjecture.
This sequence is either unbounded (in which case 44745755^4 is in fact a counterexample) or periodic.
a(229) <= 3034663491871541. - Chai Wah Wu, Aug 09 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..228
M. Filaseta et al., On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.

Cf. A020639, A076336, A213353, A258091, A336943.

A336943 Least prime factor of 44745755^4 + 2^(4n+2).

Original entry on oeis.org

797, 37, 13, 113, 29, 13, 73, 2593, 13, 41, 37, 13, 509, 57881, 13, 73, 293, 13, 29, 37, 13, 7555049, 53, 13, 41, 29, 13, 677, 37, 13, 8557781, 113, 13, 73, 41, 13, 397, 37, 13, 29, 1217, 13, 73, 9820301, 13, 113, 29, 13, 53, 41, 13, 73, 113, 13, 41, 37, 13

Offset: 0

Jeppe Stig Nielsen, Aug 08 2020

nonn

k = 44745755^4 has the property that k + 2^m is composite for all m. However, it is conjectured that this sequence is unbounded. This is the case if and only if A336347 is unbounded; because a full covering set for k*2^m + 1 would also be a full covering for k + 2^m, and vice versa.

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..500
M. Filaseta et al., On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, Journal of Number Theory, Volume 128, Issue 7 (July 2008), pp. 1916-1940.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.

Cf. A020639, A076336, A213353, A336347.

a(n) = vecmin(factor(44745755^4+2^(4*n+2))[,1]); \\ Michel Marcus, Aug 08 2020

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.

cp's OEIS Frontend

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

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A244565 Odd integers n such that for every integer k>0, n*2^k+1 has a divisor in the set { 3, 5, 7, 13, 37, 73, 109 }.

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A251057 Odd numbers n not congruent to 5 mod 6 such that for all k >= 1 the numbers n*4^k + 1 are composite.

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A260349 a(n) = min(k : A046067((k+1)/2) = n).

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A263560 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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A336347 Least prime factor of 44745755^4*2^(4n+2) + 1.

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A336943 Least prime factor of 44745755^4 + 2^(4n+2).

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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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