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.

A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 36, 42, 54, 48, 60, 84, 66, 78, 72, 126, 90, 102, 108, 114, 96, 120, 150, 138, 162, 132, 144, 168, 246, 156, 180, 186, 240, 204, 192, 216, 198, 210, 174, 258, 252, 222, 234, 228, 318, 282, 264, 276, 342, 306, 294, 312, 270, 354, 372
Offset: 1

Views

Author

Alexei Kourbatov, May 05 2020

Keywords

Comments

Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

Examples

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.

Links

Crossrefs

Cf. A007528, A014320, A058320, A268928, A330853, A334544, A334545.

Programs

  • PARI
    isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A334545(n) - A334544(n).

A334544 Primes of the form 6k - 1 preceding the first-occurrence gaps in A334543.

Original entry on oeis.org

5, 29, 113, 197, 359, 521, 1109, 1733, 4289, 6389, 7349, 8297, 9059, 12821, 35603, 37691, 58787, 59771, 97673, 105767, 130649, 148517, 153749, 180797, 220019, 328127, 402593, 406907, 416693, 542261, 780401, 1138127, 1294367, 1444271, 1463621, 1604753
Offset: 1

Views

Author

Alexei Kourbatov, May 05 2020

Keywords

Comments

Subsequence of A007528. Contains A268929 as a subsequence. First differs from A268929 at a(5)=359.
A334543 lists the corresponding gap sizes; see more comments there.

Examples

			The first two primes of the form 6k-1 are 5 and 11; we have a(1)=5. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap size 41-29=12 has not occurred before, so a(2)=29.

Links

Crossrefs

Cf. A007528, A014320, A058320, A268929, A330854, A334543, A334545.

Programs

  • PARI
    isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(s", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A334545(n) - A334543(n).

A268930 Primes 6k - 1 at the end of the maximal gaps in A268928.

Original entry on oeis.org

11, 41, 131, 227, 557, 1151, 1787, 6449, 7433, 35729, 148667, 180959, 402761, 407153, 2339297, 5522039, 11158331, 20831621, 22441073, 27681671, 73452191, 241563941, 953758661, 1444258271, 1917281867, 6822754391, 15867287129, 28265030429, 40841580683, 177858260357
Offset: 1

Views

Author

Alexei Kourbatov, Feb 15 2016

Keywords

Comments

Subsequence of A007528 and A334545.
A268928 lists the corresponding record gap sizes. See more comments there.

Examples

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 are not records so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 is a new record, so a(2)=41.

Links

Crossrefs

Cf. A007528, A268928, A268929, A334543, A334545.

Programs

  • PARI
    re=0; s=5; forprime(p=11, 1e8, if(p%6!=5, next); g=p-s; if(g>re, re=g; print1(p", ")); s=p)

Formula

a(n) = A268928(n) + A268929(n). - Alexei Kourbatov, Jun 15 2020.
Showing 1-3 of 3 results.

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