cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Altug Alkan, Mar 28 2016

Keywords

Comments

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.

Examples

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

Crossrefs

Cf. A076336, A270971.

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

Views

Author

Altug Alkan, Mar 28 2016

Keywords

Comments

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?

Examples

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

Links

Crossrefs

Cf. A076336, A244561, A270971, A270993.

Programs

  • Magma
    [9454129 + 11184810*n: n in [0..30]]; // Vincenzo Librandi, Mar 29 2016
  • Maple
    A270994:=n->9454129 + 11184810*n: seq(A270994(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016
  • Mathematica
    Table[9454129 + 11184810*n, {n, 0, 100}] (* G. C. Greubel, Mar 28 2016 *)
  • PARI
    a(n) = 9454129 + 11184810*n;
  • PARI
    x='x+O('x^99); Vec((9454129+1730681*x)/(1-x)^2)
  • Python
    for n in range(0,100):print(9454129+11184810*n) # Soumil Mandal, Apr 03 2016

Formula

G.f.: (9454129 + 1730681*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

Views

Author

Altug Alkan, Mar 30 2016

Keywords

Comments

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?

Examples

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

Links

Crossrefs

Cf. A076336, A270971, A270993.

Programs

  • Mathematica
    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 *)
  • PARI
    lista(nn) = for(n=0, nn, if(ispseudoprime(s=7523267 + 11184810*n) && nextprime(s+1) == (s+14), print1(n, ", ")));
  • PARI
    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
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.