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.

A023189 Conjecturally, number of infinitely recurring prime patterns of width 2n-1.

Original entry on oeis.org

1, 1, 1, 3, 4, 4, 14, 13, 16, 48, 55, 50, 173, 148, 147, 665, 580, 559, 1920, 1447, 1975, 6240, 4228, 5689, 15764, 17562, 14332, 46207, 39071, 35317, 172311, 134752, 110758, 381384, 299971, 479935, 1154568, 733900, 1027967, 2581763, 2636545, 2333308, 8369027, 5516720, 6043194
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Of the patterns counted by A023192, the number of those that start and end with a prime. - Sean A. Irvine, May 27 2019

Examples

			From _Jon E. Schoenfield_, May 17 2024: (Start)
The table below lists every (conjecturally) infinitely recurring prime pattern of width 2n-1 for n = 1..7. Each p represents a prime; each c represents a composite.
.
  n  2n-1  a(n)  prime patterns
  -  ----  ----  --------------------------------------------------
  1     1     1  p
  2     3     1  pcp
  3     5     1  pcccp
  4     7     3  pcccccp, pcpcccp, pcccpcp
  5     9     4  pcccccccp, pcpcccccp, pcccccpcp, pcpcccpcp
  6    11     4  pcccccccccp, pcccpcccccp, pcccccpcccp, pcccpcpcccp
  7    13    14  pcccccccccccp, pcpcccccccccp, pcccpcccccccp,
                 pcccccpcccccp, pcccccccpcccp, pcccccccccpcp,
                 pcpcccpcccccp, pcpcccccpcccp, pcccpcpcccccp,
                 pcccpcccccpcp, pcccccpcpcccp, pcccccpcccpcp,
                 pcpcccpcpcccp, pcccpcpcccpcp
(End)

Links

Crossrefs

Cf. A023190, A023191, A023192.

Extensions

Name edited by Jon E. Schoenfield, May 17 2024
a(43)-a(45) from Pontus von Brömssen, Aug 25 2025

A023191 Conjecturally, number of maximal infinitely-recurring prime patterns of width 2*n-1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 4, 2, 10, 14, 3, 2, 4, 2, 1, 2, 11, 2, 2, 6, 9, 6, 2, 12, 14, 4, 2, 2, 10, 2, 4, 8, 2, 12, 8, 4, 12, 2, 2, 2, 2, 6, 4, 20, 2, 24, 14, 4, 4, 2, 6, 92, 18, 8, 2, 18, 12, 8, 32, 8, 10, 10, 2, 12, 30, 2, 10, 2, 70, 46, 4, 4, 88, 14, 30, 20
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

See explanation in A023190. - Sean A. Irvine, May 27 2019

Links

Extensions

More terms from Thomas J Engelsma web page added by Martin Raab, Oct 31 2021

A086153 Special prime numbers arranged in a triangle: n-th row contains m primes p (where m = pi(2n + A020483(n)) - pi(A020483(n))) with following properties.

Original entry on oeis.org

3, 7, 3, 23, 5, 89, 23, 3, 139, 19, 7, 3, 199, 47, 17, 5, 113, 83, 23, 17, 3, 1831, 211, 43, 13, 7, 3, 523, 109, 79, 19, 11, 5, 887, 317, 107, 47, 17, 11, 3, 1129, 619, 109, 79, 19, 7, 1669, 199, 113, 73, 43, 13, 5, 2477, 1373, 197, 113, 71, 41, 11, 3, 2971, 1123, 199, 109
Offset: 1

Views

Author

Labos Elemer, Aug 08 2003

Keywords

Comments

1: q = p + 2n is also a prime, although not necessarily the next after p;
2: the k-th position of the n-th row gives is a prime p such that the number of further primes between p and q = p + 2n (not counting p and q) is k-1;
3: the primes p are the smallest with these properties.
Thus each row only contains primes. The first term in the n-th row is A000230(n). The last one in the same row is A020483(n). The length of the n-th row is pi(2n + A020483(n)) - pi(A020483(n)).
From Martin Raab, Aug 29 2021: (Start)
T(n,k) is zero if there is no admissible pattern with k+1 primes for the interval of length 2n under the given properties.
T(38,16) > 2^48. It requires a pattern of 17 primes with a difference of 76 between the first and the last prime. Admissible patterns of this kind exist, but solutions with 17 primes are rather hard to find. (End)
The next unknown values are T(43,19) and T(44,19), which require intervals of 20 primes with a diameter of 86 and 88, respectively. - Brian Kehrig, Jun 25 2024

Examples

			The table begins as follows:
    3;
    7,  3;
   23,  5;
   89, 23,  3;
  139, 19,  7,  3;
  199, 47, 17,  5;
  113, 83, 23, 17,  3;
  ...
For example, suppose n = 50: d = 2n = 100; the 50th row consists of 25 terms as follows: {396733, 58789, 142993, 38461, 37699, 7351, 5881, 1327, 2557, 1879, 1621, 1117, 463, 457, 283, 331, 211, 127, 73, 67, 31, ?, ?, 7, 3};
A000230(50)=396733, A020483(50)=3; between 143093 and 142993 two primes {143053,143063} occur because 142993 is the 3rd (from 2+1) entry in the 50th row.
The length of 50th row is pi(100+3) - pi(3) = pi(103) - pi(3) = 27 - 2 = 25, number of primes between 103 and 3 is 24 (not counting 103 and 3).

Links

Crossrefs

Cf. A000720, A000230, A020483, A023190, A086155, A086138-A086149, A086155.

Programs

  • Mathematica
    (* Program to generate the 19th row *) cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=38, k=0, mxc=Ceiling[d/3]; vg=PrimePi[30593]} t=Table[0, {mxc}]; t1=Table[0, {mxc}]; Do[s=cp[1+Prime[n], Prime[n]+d-1]; np=d+Prime[n]; If[PrimeQ[np]&&s<(1+mxc)&&t[[s+1]]==0, t[[s+1]]=n; t1[[s+1]]=Prime[n]], {n, 1, 5000}]; {t, t1}
  • PARI
    {z=concat(vector(13),binary(8683781)); for(n=1, 37, p1=3; while(!isprime(p1+2*n), p1=nextprime(p1+2)); p2=p1+2*n; k=primepi(p2)-primepi(p1); r=vector(k); r[k]=p1; i1=1; i2=0; s=vecsort(r); while(s[1+z[n]]==0, while(i1*i2==0, p1+=2; p2+=2; i2=isprime(p2); k=k-i1+i2; i1=isprime(p1)); if(!r[k], r[k]=p1; s=vecsort(r)); i2=0); print("row "n": "r))} \\ Martin Raab, Oct 21 2021
Showing 1-3 of 3 results.

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