A244071 -id:A244071 - 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.

A244561 Odd integers m such that for every integer k > 0, m*2^k+1 has a divisor in the set {3, 5, 7, 13, 17, 241}.

Original entry on oeis.org

271129, 271577, 482719, 575041, 603713, 903983, 965431, 1518781, 1624097, 1639459, 2131043, 2131099, 2541601, 2931767, 2931991, 3083723, 3098059, 3555593, 3608251, 4067003, 4573999, 6134663, 6135559, 6557843, 6676921, 6678713, 6742487, 6799831, 7400371, 7523267, 7523281, 7761437, 7765021, 7892569, 8007257, 8629967, 8840599, 8871323, 9208337, 9454129, 9454157, 9854491, 9854603, 9930469, 9937637, 10192733, 10422109, 10675607

Offset: 1

Pierre CAMI, Jun 30 2014

nonn

For n > 48, a(n) = a(n-48) + 11184810; the first 48 values are in the data.
The set {3, 5, 7, 13, 17, 241} is the set of prime divisors of 2^24 - 1. Hence for every p in the set the multiplicative order of 2 modulo p divides 24. Note that twice the product of {3, 5, 7, 13, 17, 241} is 11184810. - Jeppe Stig Nielsen, Mar 10 2019
Subset of provable Sierpiński numbers A076336. - Jeppe Stig Nielsen, Mar 10 2019

Cf. A076336, A244070, A244071, A244072, A244073, A244074, A244076.

D=[3, 5, 7, 13, 17, 241];P=2*lcm(D);M=lcm(apply(d->znorder(Mod(2,d)),D));forstep(k=1,+oo,2,if(k%P==1,print();print());for(n=0,M-1,for(i=1,#D,k*Mod(2,D[i])^n+1==0 && next(2));next(2));print1(k,", ")) \\ Jeppe Stig Nielsen, Mar 10 2019

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

A244562 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, 73 }.

Original entry on oeis.org

78557, 2191531, 2510177, 2576089, 7134623, 7696009, 8184977, 10275229, 10391933, 11201161, 12151397, 12384413, 12756019, 13065289, 13085029, 15168739, 16391273, 18140153, 18156631, 19436611, 19558853, 20312899, 20778931, 21610427

Offset: 1

Pierre CAMI, Jun 30 2014

nonn

For n > 144 a(n) = a(n-144) + 140100870, the first 144 values are in the table.

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

Cf. A076336, A244071, A244561.

For n > 144 a(n) = a(n-144) + 140100870.

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.

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

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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A244561 Odd integers m such that for every integer k > 0, m*2^k+1 has a divisor in the set {3, 5, 7, 13, 17, 241}.

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A244562 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, 73 }.

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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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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 }.

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