A270993 -id:A270993 - cp's OEIS frontend

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

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.

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.

A271080 Integers k such that s(k) = 7523267 + 11184810*k and s(k) + 14 are consecutive primes.

Original entry on oeis.org

8, 16, 82, 101, 132, 187, 201, 253, 265, 300, 318, 351, 393, 408, 429, 449, 474, 489, 508, 660, 662, 673, 687, 772, 869, 877, 880, 924, 945, 958, 963, 984, 1028, 1042, 1070, 1083, 1124, 1134, 1226, 1249, 1257, 1265, 1319, 1340, 1345, 1352, 1365, 1389, 1463, 1664, 1816, 1834, 1878, 1969

Offset: 1

Altug Alkan, Mar 30 2016

nonn

s(k) and s(k) + 14 are always Sierpiński numbers for k >= 0.
Motivated by the question: What are the consecutive Sierpiński numbers with difference 14 that are also consecutive primes?
See A270971 and A270993 for the reason for the definition's focus on 14.
How does the graph of this sequence look for larger values of n?

			8 is a term because 7523267 + 11184810*8 = 97001747 and 97001761 are consecutive (provable) Sierpiński numbers and they are also consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A076336, A270971, A270993.

Select[Range@ 2000, And[PrimeQ@ #, NextPrime@ # == # + 14] &@(7523267 + 11184810 #) &] (* Michael De Vlieger, Mar 30 2016 *)
cpQ[n_]:=Module[{c=7523267+11184810n},PrimeQ[c]&&NextPrime[c]==c+14]; Select[Range[ 2000],cpQ] (* Harvey P. Dale, Oct 07 2023 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(s=7523267 + 11184810*n) && nextprime(s+1) == (s+14), print1(n, ", ")));

is(n)=my(s=11184810*n+7523267); isprime(s) && isprime(s+14) && !isprime(s+6) && !isprime(s+12) \\ Charles R Greathouse IV, Mar 31 2016

cp's OEIS Frontend

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

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

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Magma

Maple

Mathematica

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PARI

Python

Formula

A271080 Integers k such that s(k) = 7523267 + 11184810*k and s(k) + 14 are consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI