A076337 -id:A076337 - cp's OEIS frontend

A244070 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 17, 241}.

509203, 762701, 992077, 1247173, 1254341, 1330207, 1330319, 1730653, 1730681, 1976473, 2313487, 2344211, 2554843, 3177553, 3292241, 3419789, 3423373, 3661529, 3661543, 3784439, 4384979, 4442323, 4506097, 4507889, 4626967, 5049251, 5050147, 6610811, 7117807, 7576559, 7629217, 8086751, 8101087, 8252819, 8253043, 8643209, 9053711, 9053767, 9545351, 9560713, 9666029, 10219379, 10280827, 10581097, 10609769, 10702091, 10913233, 10913681

Offset: 1

Pierre CAMI, Jun 19 2014

nonn

For n > 48, a(n) = a(n-48) + 11184810, the first 48 values are given in the data.

Cf. A076337, A244071, A244072, A244073, A244074, A244076.

For n > 48, a(n) = a(n-48) + 11184810.

A244072 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 19, 37, 109}.

Original entry on oeis.org

2136283, 2251349, 2924861, 3781541, 8010517, 10645867, 11124703, 16413457, 16593431, 17713229, 21992527, 22146359, 23572417, 23835883, 23909173, 24359437, 24688477, 25124999, 26711801, 26880179, 27094349, 28151593, 30577271, 32126257

Offset: 1

Pierre CAMI, Jun 19 2014

nonn

For n > 144 a(n) = a(n-144) + 209191710, the first 144 values are in the table.

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

Cf. A076337, A244070, A244071, A244073, A244074, A244076.

For n > 144, a(n) = a(n-144) + 209191710.

A244073 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 19, 73, 109}.

Original entry on oeis.org

1744117, 6975809, 7790113, 11942443, 13006807, 16861093, 16882181, 17207051, 20003369, 20147891, 21013423, 25638127, 42918821, 45113083, 47285977, 48635609, 49884041, 53335151, 53538727, 56592041, 63412693, 63750101, 64062209, 65739209

Offset: 1

Pierre CAMI, Jun 19 2014

nonn

For n > 144 a(n) = a(n-144) + 412729590, the first 144 values are in the table.

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

Cf. A076337, A244070, A244071, A244072, A244074, A244076.

For n > 144, a(n) = a(n-144) + 412729590.

A244074 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

1830187, 4643293, 17041931, 20787701, 50462309, 52363777, 66659587, 68026001, 71604733, 71817943, 88558303, 91609361, 93193151, 97363751, 118421557, 122606647, 123765359, 124808009, 131118733, 131408411, 134320001, 135411719, 139778591, 142339723

Offset: 1

Pierre CAMI, Jun 19 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. A076337, A244070, A244071, A244072, A244073, A244076.

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

A244076 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 17, 97, 257}.

Original entry on oeis.org

10157893, 71627707, 86571727, 90109601, 98957849, 99023257, 99284501, 114096371, 142399363, 166262293, 207549337, 213185347, 213708611, 215798563, 229298773, 229306949, 242872709, 251719903, 274263943, 276356999, 278326889, 284716807, 289074853, 294317669

Offset: 1

Pierre CAMI, Jun 19 2014

nonn

For n > 96 a(n) = a(n-96) + 1156954890, the first 96 values are in the table.

Pierre CAMI, Table of n, a(n) for n = 1..96

Cf. A076337, A244070, A244071, A244072, A244073, A244074.

For n > 96 a(n) = a(n-96) + 1156954890.

A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, 1, 4, 1, 1, 2, 1, 1, 12, 3, 2, 4, 5, 1, 2, 7, 1, 2, 1, 3, 2, 5, 1, 4, 1, 3, 2, 1, 1, 10, 3, 2, 10, 9, 2, 8, 1, 1, 12, 1, 2, 2, 25, 1, 2, 3, 1, 2, 1, 1, 2, 5, 1, 4, 5, 3, 2, 1, 1, 2, 3, 2, 4, 1, 2, 2, 1, 1, 8, 3, 4, 2, 1, 3, 226, 3, 1, 2, 1, 1, 2

Offset: 1

Jorge Coveiro, Jun 04 2005

nonn

It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
Browkin & Schinzel, having proved that 509203*2^k - 1 is composite for all k > 0, ask for the first such number with this property, noting that the question is implicit in Aigner 1961. - Charles R Greathouse IV, Jan 12 2018
Record values begin a(1) = 2, a(7) = 3, a(12) = 4, a(22) = 7, a(30) = 12, a(64) = 25, a(96) = 226, a(330) = 800516; the next record appears to be a(1147), unless a(1147) = -1. (The value for a(330), i.e., for k = 659, is from the Ballinger & Keller link, which also lists k = 2293, i.e., n = (k+1)/2 = (2293+1)/2 = 1147, as the smallest of 50 values of k < 509203 for which no prime of the form k*2^m-1 had yet been found.) - Jon E. Schoenfield, Jan 13 2018
Same as A046069 except for a(2) = 1. - Georg Fischer, Nov 03 2018

Hans Riesel, Några stora primtal, Elementa 39 (1956), pp. 258-260.

Jon E. Schoenfield, Table of n, a(n) for n = 1..329
A. Aigner, Folgen der Art ar^n + b, welche nur teilbare Zahlen liefern, Math. Nachr. 23 (1961), pp. 259-264. (Cited in Browkin & Schinzel)
R. Ballinger & W. Keller, The Riesel Problem: Definition and Status.
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), pp. 55-58.
Wilfrid Keller, List of primes k.2^n - 1 for k < 300 .
Hans Riesel, Some large prime numbers. Translated from the Swedish original (Några stora primtal, Elementa 39 (1956), pp. 258-260) by Lars Blomberg.

Main sequences for Riesel problem: A038699, A040081, A046069, A050412, A052333, A076337, A101036, A108129.

Cf. A040081, A046069.

Array[Function[k, SelectFirst[Range@300, PrimeQ[k 2^# - 1] &]][2 # - 1] &, 102] (* Michael De Vlieger, Jan 12 2018 *)
smk[n_]:=Module[{m=1,k=2n-1},While[!PrimeQ[k 2^m-1],m++];m]; Array[smk,120] (* Harvey P. Dale, Dec 26 2023 *)

forstep(k=1,301,2,n=1;while(!isprime(k*2^n-1),n++);print1(n,","))

Edited by Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 25 2006

Name corrected by T. D. Noe, Feb 13 2011

A243969 Integers n not of form 3m+2 such that for any integer k > 0, n*10^k+1 has a divisor in the set { 7, 11, 13, 37 }.

Original entry on oeis.org

9175, 9351, 17676, 24826, 26038, 28612, 38026, 38158, 46212, 46927, 48247, 56473, 61863, 63075, 63898, 65649, 75063, 75195, 83425, 83964, 85284, 91750, 93510, 100935

Offset: 1

Pierre CAMI, Jun 16 2014

nonn

For n>24 a(n) = a(n-24) + 111111, the first 24 values are in the data.

If n is of form 3m+2 then n*10^k+1 is always divisible by 3. The sequence is a base 10 variant of provable Sierpiński numbers (A076336). It is currently unknown whether 7666*10^k+1 is always composite but based on heuristics it probably has large undiscovered primes. 7666 is the only remaining base 10 Sierpiński candidate below 9175. - Jens Kruse Andersen, Jul 09 2014

			9175*10^k+1 is divisible by 11 for k of form 6m+1, 6m+3, 6m+5, by 37 for k of form 6m (and also 6m+3), by 13 for 6m+2, and by 7 for 6m+4. This covers all k. {7, 11, 13, 37} is called a covering set. - _Jens Kruse Andersen_, Jul 09 2014

A. Brunner, C. Caldwell, D. Krywaruczensko, C. Lownsdale, Generalized Sierpiński Numbers Base b (has a typo in covering set for 9175, base 10. - _Jens Kruse Andersen_, Jul 09 2014)

Cf. A076336, A076337, A243974, A244070.

For n>24 a(n) = a(n-24) + 111111.

Definition corrected by Jens Kruse Andersen, Jul 09 2014

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

Arkadiusz Wesolowski, Jun 13 2012

nonn

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.

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

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary, Riesel number

Cf. A076337, A101036, A187714, A222534.

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

A243974 Integers n not of form 3m+1 such that for any integer k>0, n*10^k-1 has a divisor in the set { 7, 11, 13, 37 }.

Original entry on oeis.org

10176, 17601, 19361, 25827, 27147, 27686, 35916, 36048, 45462, 47213, 48036, 49248, 54638, 62864, 64184, 64899, 72953, 73085, 82499, 85073, 86285, 93435, 101760, 101936

Offset: 1

Pierre CAMI, Jun 16 2014

nonn

For n>24 a(n) = a(n-24) + 111111, the first 24 values are in the data.

If n is of form 3m+1 then n*10^k-1 is always divisible by 3. - Jens Kruse Andersen, Jul 09 2014

			10176*10^k-1 is divisible by 11 for k of form 6m, 6m+2, 6m+4, by 7 for k of form 6m+1, by 37 for 6m+3 (and also 6m), and by 13 for 6m+5. This covers all k. {7, 11, 13, 37} is called a covering set. - _Jens Kruse Andersen_, Jul 09 2014

Cf. A076337, A243969, A243974, A244070, A244071, A244072, A244073, A244074, A244076.

For n > 24, a(n) = a(n-24) + 111111.

Definition corrected by Jens Kruse Andersen, Jul 09 2014

A244211 Integers n such that for every integer k>0, n*6^k-1 has a divisor in the set { 7, 13, 31, 37, 43 }.

Original entry on oeis.org

133946, 213410, 299144, 33845, 367256, 803676, 1214450, 1250446, 1280460, 1704478, 1780150, 1792762, 1794864, 2003070, 2004962, 2203536, 2798489, 3014465, 3027709, 3041998, 3053350, 3194549, 3326301, 4244794

Offset: 1

Pierre CAMI, Jun 23 2014

nonn

For n > 24 a(n) = a(n-24) + 4488211, the first 24 values are in the data.

When the number a(n) has 1 or 6 as the last digit, the number a(n)*6^k-1 is always divisible by 5 and always has another divisor in the set { 7, 13, 31, 37, 97 } for every k.

Cf. A076337, A243969, A244070, A244071, A244072, A244073, A244074, A244076.

For n > 24, a(n) = a(n-24) + 4488211.

cp's OEIS Frontend

A244070 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 17, 241}.

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A244072 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 19, 37, 109}.

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A244073 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 19, 73, 109}.

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A244074 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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A244076 Odd integers n such that for every integer k>0, n*2^k-1 has a divisor in the set {3, 5, 7, 13, 17, 97, 257}.

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A108129 Riesel problem: let k=2n-1; then a(n)=smallest m >= 1 such that k*2^m-1 is prime, or -1 if no such prime exists.

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A243969 Integers n not of form 3m+2 such that for any integer k > 0, n*10^k+1 has a divisor in the set { 7, 11, 13, 37 }.

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

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A243974 Integers n not of form 3m+1 such that for any integer k>0, n*10^k-1 has a divisor in the set { 7, 11, 13, 37 }.

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