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.

Previous Showing 11-16 of 16 results.

A330855 Primes 6k + 1 at the end of first-occurrence gaps in A330853.

Original entry on oeis.org

13, 31, 61, 271, 307, 1381, 1531, 1987, 2437, 4423, 7867, 10243, 16831, 22273, 24337, 38557, 40351, 43543, 69661, 75511, 100927, 119047, 171403, 195691, 204301, 250423, 480343, 577807, 590593, 1164799, 1207903, 1278997, 1382419, 1468189, 1526929, 1890019, 2314591
Offset: 1

Views

Author

Alexei Kourbatov, Apr 27 2020

Keywords

Comments

Subsequence of A002476. Contains A268927 as a subsequence. First differs from A268927 at a(5)=307.
A330853 lists the corresponding gap sizes; see more comments there.

Examples

			The first two primes of the form 6k+1 are 7 and 13, so a(1)=13. The next prime 6k+1 is 19, and the gap 19-13=6 already occurred, so a new term is not added to the sequence. The next prime 6k+1 is 31, and the gap 31-19=12 is occurring for the first time; therefore a(2)=31.

Links

Crossrefs

Cf. A002476, A014320, A058320, A268927, A330853 (first-occurrence gap sizes), A330854 (primes beginning the first-occurrence gaps).

Programs

  • PARI
    isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(p", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A330853(n) + A330854(n).

A014321 The next new gap between successive odd primes (divided by 2).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 6, 9, 10, 11, 17, 12, 8, 13, 14, 15, 16, 18, 22, 21, 20, 26, 24, 19, 36, 25, 31, 27, 30, 29, 23, 28, 32, 34, 43, 33, 35, 39, 38, 41, 48, 56, 50, 37, 45, 42, 57, 40, 44, 49, 46, 53, 47, 59, 66, 52, 51, 55, 63, 60, 74, 54, 61, 69, 64, 77, 65, 58, 73, 68, 62, 67
Offset: 1

Views

Author

Hynek Mlcousek (hynek(AT)dior.ics.muni.cz)

Keywords

Comments

If Polignac's conjecture holds (which is highly likely), then this sequence is a permutation of the positive integers. Even a weaker form of the conjecture would be enough: "Every even number occurs at least once as difference of subsequent primes". - Ferenc Adorjan (ferencadorjan(AT)gmail.com), May 17 2007

Links

Crossrefs

Cf. A014320.
Equals A058320(n+1)/2.
Inverse: A130264, Cf. A086979.

Programs

  • Mathematica
    DeleteDuplicates[Differences[Prime[Range[2,500000]]]]/2 (* Harvey P. Dale, Sep 15 2023 *)

Extensions

More terms from Ferenc Adorjan (ferencadorjan(AT)gmail.com), May 17 2007

A334545 Primes of the form 6k - 1 at the end of first-occurrence gaps in A334543.

Original entry on oeis.org

11, 41, 131, 227, 383, 557, 1151, 1787, 4337, 6449, 7433, 8363, 9137, 12893, 35729, 37781, 58889, 59879, 97787, 105863, 130769, 148667, 153887, 180959, 220151, 328271, 402761, 407153, 416849, 542441, 780587, 1138367, 1294571, 1444463, 1463837, 1604951
Offset: 1

Views

Author

Alexei Kourbatov, May 05 2020

Keywords

Comments

Subsequence of A007528. Contains A268930 as a subsequence. First differs from A268930 at a(5)=383.
A334543 lists the corresponding gap sizes; see more comments there.

Examples

			The first two primes of the form 6k-1 are 5 and 11, so we have a(1)=11. 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)=41.

Links

Crossrefs

Cf. A007528, A014320, A058320, A268930, A330855, A334543, A334544.

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(p", "); isFirstOcc[g/6]=0); s=p)

Formula

a(n) = A334543(n) + A334544(n).

A057700 The next new gap between successive lucky numbers.

Original entry on oeis.org

2, 4, 6, 12, 8, 10, 18, 16, 24, 14, 22, 20, 30, 28, 26, 52, 32, 36, 34, 38, 46, 42, 40, 64, 44, 50, 54, 48, 56, 58, 66, 68, 60, 62, 100, 78, 72, 94, 76, 84, 96, 82, 74, 108, 70, 90, 80, 92, 86, 112, 144, 132, 88, 120, 106, 122, 140, 102, 104, 114, 110, 98, 158
Offset: 0

Views

Author

Naohiro Nomoto, Oct 23 2000

Keywords

Crossrefs

Cf. A000959, A014320, A031883.

Extensions

More terms from Sascha Kurz, Mar 24 2002
a(51) onward corrected by Sean A. Irvine, Jun 23 2022

A090811 Next new gap between successive prime powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 9, 20, 7, 18, 22, 30, 16, 26, 28, 24, 32, 17, 34, 36, 44, 42, 27, 40, 52, 48, 38, 72, 50, 62, 54, 60, 58, 15, 46, 56, 64, 29, 68, 86, 66, 70, 78, 76, 82, 96, 112, 100, 74, 90, 84, 114, 21, 80, 88, 98, 92
Offset: 1

Views

Author

Naohiro Nomoto, Feb 11 2004

Keywords

Crossrefs

Cf. A000961, A014320.

Programs

  • Mathematica
    DeleteDuplicates[Differences[Select[Range[1000000],PrimePowerQ]]] (* Harvey P. Dale, Oct 21 2018 *)

A356763 Triprime gaps (A114403) in the order of first occurrence.

Original entry on oeis.org

4, 6, 2, 7, 1, 12, 5, 11, 3, 14, 8, 9, 10, 18, 13, 15, 16, 21, 17, 19, 22, 32, 24, 20, 23, 29, 28, 25, 26, 33, 34, 27, 30, 31, 37, 40, 35, 36, 46, 39, 41, 44, 45, 42, 38, 50, 58, 43, 51, 54, 49, 52, 48, 47, 56, 55, 53, 60, 57, 59, 63, 61, 65, 66, 69, 64, 62, 67, 68, 70, 83, 71, 73, 78, 72
Offset: 1

Views

Author

Zak Seidov, Aug 26 2022

Keywords

Crossrefs

Cf. A114403, A014612.
Cf. A014320 (for prime gaps), A356769 (for semiprime gaps).

Programs

  • Mathematica
    DeleteDuplicates[Differences[Select[Range[10^6], PrimeOmega[#] == 3 &]]] (* Amiram Eldar, Aug 26 2022 *)
Previous Showing 11-16 of 16 results.

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