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.

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A078849 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[2, 6,6]; short d-string notation of pattern = [266].

Original entry on oeis.org

149, 599, 3299, 4649, 5099, 6359, 11489, 12539, 16979, 19469, 27059, 30089, 31319, 34259, 42179, 53609, 58229, 63689, 65699, 71339, 75209, 77549, 78569, 80909, 81929, 85829, 87509, 87539, 89519, 92219, 101279, 105359, 112289, 116099, 116789
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A049437. - R. J. Mathar, Feb 10 2013

Examples

			149, 149+2=151, 149+2+6=157, 149+2+6+6=163 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    d = {2, 6, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)
Select[Partition[Prime[Range[12000]],4,1],Differences[#]=={2,6,6}&][[All,1]] (* Harvey P. Dale, Dec 29 2017 *)

Formula

Primes p = p(i) such that p(i+1)=p+2, p(i+2)=p+2+6, p(i+3)=p+2+6+6.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009

A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

Original entry on oeis.org

67, 1447, 2377, 2707, 5437, 5737, 7207, 9337, 11827, 12037, 19207, 21487, 21517, 23197, 26107, 26947, 28657, 31147, 31177, 35797, 37357, 37567, 42697, 50587, 52177, 65167, 67927, 69997, 71707, 74197, 79147, 81547, 103087, 103387, 106657
Offset: 1

Views

Author

Labos Elemer, Dec 11 2002

Keywords

Comments

Subsequence of A022005. - R. J. Mathar, May 06 2017

Examples

			p=67,67+4=71,67+4+2=73,67+4+2+6=79 are consecutive primes.

Links

Crossrefs

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

Programs

  • Mathematica
    d = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)

Formula

Primes p = p(i) such that p(i+1)=p+4, p(i+2)=p+4+2, p(i+3)=p+4+2+6.

Extensions

Listed terms verified by Ray Chandler, Apr 20 2009

A079016 Suppose p and q = p+12 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601
Offset: 1

Views

Author

Labos Elemer, Jan 24 2003

Keywords

Examples

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

Crossrefs

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.
Cf. A079017-A079024.

Programs

  • Mathematica
    Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A078958 Primes p such that the differences between the 5 consecutive primes starting with p are (6,2,4,6).

Original entry on oeis.org

1601, 3911, 12101, 14621, 32051, 68891, 122021, 191441, 258101, 259151, 276581, 278801, 305471, 347051, 390101, 394721, 418331, 419591, 421691, 470201, 482501, 509681, 678641, 683471, 832361, 844421, 914351, 929051, 977351, 997091, 1043831, 1074701, 1104731, 1224851
Offset: 1

Views

Author

Labos Elemer, Dec 19 2002

Keywords

Comments

Equivalently, primes p such that p, p+6, p+8, p+12 and p+18 are consecutive primes.

Examples

			3911 is in the sequence since 3911, 3917 = 3911 + 6, 3919 = 3911 + 8, 3923 = 3911 + 12 and 3929 = 3911 + 18 are consecutive primes.

Links

Crossrefs

Subsequence of A078853. - R. J. Mathar, May 06 2017
Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Programs

  • Mathematica
    Select[Partition[Prime[Range[82000]],5,1],Differences[#]=={6,2,4,6}&][[All,1]] (* Harvey P. Dale, Jul 09 2021 *)
  • PARI
    list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 2 && p4 - p3 == 4 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

Formula

a(n) == 11 (mod 30). - Amiram Eldar, Feb 22 2025

Extensions

Edited by Dean Hickerson, Dec 20 2002
Previous Showing 11-14 of 14 results.

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