A244561 -id:A244561 - cp's OEIS frontend

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 }.

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.

A244563 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

1290677, 4095859, 5841947, 7158107, 8959163, 9044629, 9252323, 9933857, 10306187, 11000303, 15598231, 16010419, 16625747, 16907749, 18068693, 19428919, 20189993, 23487497, 25614893, 26471633, 28410121, 30375901, 30666137, 32552687

Offset: 1

Pierre CAMI, Jun 30 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. A076336, A244072, A244561, A244562.

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

A244564 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

934909, 1259779, 6828631, 11822359, 12151397, 15285707, 17220887, 23277113, 25912463, 32971909, 34689511, 38206517, 38257411, 45181667, 46337843, 48339497, 57410477, 63676073, 67510217, 68468753, 68708387, 69169397, 70312793, 71151293

Offset: 1

Pierre CAMI, Jun 30 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. A076336, A244073, A244561, A244562, A244563.

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

A270993 Values of A076336(n) such that A076336(n) = A076336(n+1) - 14.

Original entry on oeis.org

7523267, 18708077, 29892887, 41077697, 52262507, 63447317, 74632127, 85816937, 97001747, 108186557, 119371367, 130556177, 141740987, 152925797, 164110607, 175295417, 186480227, 197665037, 208849847, 220034657, 231219467, 242404277, 253589087, 264773897, 275958707, 287143517, 298328327, 309513137

Offset: 1

Altug Alkan, Mar 28 2016

nonn

See A270971 for the motivation behind this sequence.
Riesel showed that there are infinitely many integers such that k*(2^m) - 1 is not prime for any integer m. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.
In this sequence, the lesser of (provable) Sierpiński pairs appears with the linear formula a(n) = 7523267 + 11184810*(n-1).
Since 7523267 is a term of A244561, for every integer k > 0, 7523267*2^k+1 has a divisor in the set {3, 5, 7, 13, 17, 241}. Because 11184810 = 2*3*5*7*13*17*241, a(n)*2^k+1 = 7523267*2^k+1 + 11184810*(n-1)*2^k+1 always has a divisor in the set {3, 5, 7, 13, 17, 241}. Since a(n) is always odd because of its definition, a(n) is a Sierpiński number. Additionally, 7523267 + 14 = 7523281 is also a term of A244561. So a(n) + 14 is a Sierpiński number too, with the same proof.
In conclusion, if the minimum difference between consecutive (provable) Sierpiński numbers is 14 (see comment section of A270971 for the reason behind this claim), a(n) and a(n) + 14 must be consecutive and a(n) = 7523267 + 11184810*(n-1) is the formula for this sequence.

			7523267 is a term because 7523267 and 7523267 + 14 = 7523281 are consecutive (provable) Sierpiński numbers.

Cf. A076336, A270971.

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.

A244566 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

327739, 5455789, 8879993, 9043831, 21823667, 25763447, 29949559, 75037639, 92732027, 119863547, 119879899, 122091961, 146880319, 151060223, 152106751, 163378771, 181339441, 182384417, 182646049, 218039041, 232190537

Offset: 1

Pierre CAMI, Jun 30 2014

These are the Sierpiński numbers (A076336) with covering set {3, 5, 7, 13, 17, 97, 257}. - David W. Wilson, Jul 18 2014

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. A076336, A244076, A244561, A244562, A244563, A244564, A244565.

is(n)=my(G=578477445,t=Mod(n,G)); for(k=1,768,t*=2; if(gcd(t+1, G)==1, return(0))); n%2 \\ Charles R Greathouse IV, Jul 18 2014

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

A270994 a(n) = 9454129 + 11184810*n.

Original entry on oeis.org

9454129, 20638939, 31823749, 43008559, 54193369, 65378179, 76562989, 87747799, 98932609, 110117419, 121302229, 132487039, 143671849, 154856659, 166041469, 177226279, 188411089, 199595899, 210780709, 221965519, 233150329, 244335139, 255519949, 266704759, 277889569, 289074379, 300259189

Offset: 0

Altug Alkan, Mar 28 2016

See A270971 for the motivation.
These are all Sierpiński numbers.
Since 9454129 is a term of A244561, for every integer k > 0, 9454129*2^k + 1 has a divisor in the set {3, 5, 7, 13, 17, 241}. And because 11184810 = 2*3*5*7*13*17*241, a(n)*2^k + 1 = 9454129*2^k + 1 + 11184810*n*2^k + 1 always has a divisor in the set {3, 5, 7, 13, 17, 241}. Since a(n) is always odd because of its definition, a(n) is a Sierpiński number.
Also 9454129 + 28 = 9454157 is a term of A244561. So, with the same proof, a(n) + 28 is a Sierpiński number too.
Are a(n) and a(n) + 28 always consecutive Sierpiński numbers?

			a(1) = 9454129 + 11184810*1 = 20638939.

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

Cf. A076336, A244561, A270971, A270993.

[9454129 + 11184810*n: n in [0..30]]; // Vincenzo Librandi, Mar 29 2016

A270994:=n->9454129 + 11184810*n: seq(A270994(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016

Table[9454129 + 11184810*n, {n, 0, 100}] (* G. C. Greubel, Mar 28 2016 *)

PARI
```
a(n) = 9454129 + 11184810*n;
```

x='x+O('x^99); Vec((9454129+1730681*x)/(1-x)^2)

for n in range(0,100):print(9454129+11184810*n) # Soumil Mandal, Apr 03 2016

G.f.: (9454129 + 1730681*x)/(1 - x)^2.

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

cp's OEIS Frontend

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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A244563 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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A244564 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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A270993 Values of A076336(n) such that A076336(n) = A076336(n+1) - 14.

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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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A244566 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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A270994 a(n) = 9454129 + 11184810*n.

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