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 1-2 of 2 results.

A186311 Least k such that the interval 100k to 100k+99 has exactly n primes.

Original entry on oeis.org

16718, 1559, 3020, 588, 314, 188, 186, 59, 48, 41, 21, 13, 11, 19, 5, 8, 2, 4, 1228537713709, 14688670051164208, 203860951641372730864, 1
Offset: 0

Views

Author

T. D. Noe, Feb 22 2011

Keywords

Comments

It is known that a(25)=0. Terms for n = 22 and 23 are unknown. Glaisher tabulates the number of centuries having 0, 1, 2, ... primes for numbers up to 9000000. Glaisher's 1883 book is still in print!
a(24) does not exist because the only century having 24 primes is 0 to 99 -- the same century having 25 primes. From A020497, we see that a range of 101 numbers is required to find 24 primes. Dickson's conjecture implies that a(n) exists for n=18..23. - Charles R Greathouse IV, Feb 24 2011
To see that Dickson's conjecture is applicable to the preceding statement, the appropriate general sequence to consult is A364678, which affirms that 23 primes are permissible between adjacent multiples of 100, as opposed to in an arbitrary interval of 99 integers. - Peter Munn, Sep 04 2023
a(n) for n = 18..23 is greater than 10^10. Ribenboim discusses Dickson's conjecture in two books. - T. D. Noe, Feb 24 2011
a(19) <= 1108851311300675700427. - Donovan Johnson, Feb 28 2011
a(20) <= 394338677302163715754576644. - Tim Johannes Ohrtmann, Aug 27 2015

References

  • James Glaisher, Factor Table for the Sixth Million, Taylor and Francis, London, 1883.
  • Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY, 1995, p. 372.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY, 2004, p. 250.

Links

Crossrefs

Cf. A038822 (number of primes between 100n and 100n+99).
Cf. A181098 (centuries without primes).
Cf. A186393-A186408 (centuries having 1 to 16 primes).
Cf. A186509 (centuries having 17 primes).
Cf. A361723 (centuries having 18 primes).
Cf. A020497, A261571, A364678.

Programs

  • Mathematica
    t = Differences[PrimePi[100*Range[0, 20000]]]; Flatten[Table[Position[t, n, 1, 1], {n, 0, 17}] - 1]
  • PARI
    a(n)=for(k=0,9e99,if(sum(i=100*k+1,100*k+99,ispseudoprime(i))==n, return(k))) \\ Charles R Greathouse IV, Feb 24 2011

Extensions

a(18) from Donovan Johnson, Feb 28 2011
a(19) from Brian Kehrig, Apr 08 2023
a(20)-a(21) from Brian Kehrig, May 28 2024

A361723 Numbers k such that there are 18 primes between 100*k and 100*k + 99.

Original entry on oeis.org

1228537713709, 23352869714018, 28703237474266, 144785865481702, 161394923966449, 168975708209638, 174748809066898, 207552241231357, 278215179205531, 312303328909720, 592248982143877, 812939886634531, 939100782752014, 983930290209021, 1111161494544274
Offset: 1

Views

Author

Brian Kehrig, Mar 21 2023

Keywords

Comments

There are A261571(18) = 948729 possible patterns for centuries having 18 primes.

Examples

			1228537713709 is in the sequence because there are 18 primes between 122853771370900 and 122853771370999: 122853771370900 + x, where x is one of (1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, or 99).

Links

  • Brian Kehrig, Table of n, a(n) for n = 1..39 (terms up to 10^16)
  • Note (Apr 24 2024): An older version of the b-file missed a(33) and a(38). The present b-file is correct.

Crossrefs

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).
Cf. A181098 (no primes), A186393-A186408 (1 to 16 primes), A186509 (17 primes).
Cf. A261571 (number of patterns for centuries with n primes).

Programs

  • PARI
    isok(k) = sum(i=0, 99, isprime(100*k + i)) == 18; \\ Michel Marcus, Mar 23 2023
Showing 1-2 of 2 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.