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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A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

Original entry on oeis.org

121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321
Offset: 1

Views

Author

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

Keywords

Comments

For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.
All terms are congruent to 9 (mod 14). - Zak Seidov, May 03 2017
The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3*10^11, cf. A210727. [With a slope of a(n)/n ~ 5*10^8 this would correspond to n ~ 600.] This sequence consists of first members of pairs of consecutive primes in A059044. Conversely, a pair of consecutive primes in this sequence starts a CPAP-7. This must have a common difference >= 210. As of today, the smallest known CPAP-7 starts at 382003672700092872707633 ~ 3.8*10^23, cf. Andersen link. - M. F. Hasler, Oct 27 2018
The common difference of 60 first occurs at a larger-than-expected prime. The first CPAP-6 with common difference 90 starts at 8560443932347. The first CPAP-6 with common difference 120 starts at 1925601119017087. - Jerry M Lagrou, Jan 01 2024

Links

Crossrefs

Cf. A006560: first prime to start a CPAP-n.
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A052239: starting prime of first CPAP-4 with common difference 6n.
Cf. A059044: starting primes of CPAP-5.
Cf. A210727: starting primes of CPAP-5 with common difference 60.

Programs

  • PARI
    p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018

Formula

Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - M. F. Hasler, Oct 30 2018

Extensions

Corrected by Jud McCranie, Jan 04 2001
a(11)-a(18) from Donovan Johnson, Sep 05 2008
Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018

A335406 First position of n in the sequence of run-lengths of the sequence of prime gaps.

Original entry on oeis.org

1, 2, 49, 633353, 6706139
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2020

Keywords

Comments

Prime gaps are differences between adjacent prime numbers.

Crossrefs

Positions of first appearances in A333254.
The unequal version is 7, 1, 4, 15, 10, 36, 5, 6, 84, ...
The weakly decreasing version is 1, 2, 7, 23, 26, ...
The weakly increasing version is 5, 2, 3, 1, 81, 193, ...
The strictly decreasing version is 1, 4, 8, 150, 160, ...
The strictly increasing version is 6, 1, 4, 38, 221, ...
Prime gaps are A001223.
The first term of the first length-n arithmetic progression of consecutive primes is A006560(n), with index A089180(n).
Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Cf. A000040, A031217, A054800, A059044, A084758, A090832, A106356, A124767, A238279, A295235, A006560.

Programs

  • Mathematica
    qe=Length/@Split[Differences[Array[Prime,10000]],SameQ];
Table[Position[qe,i][[1,1]],{i,Union[qe]}]

Extensions

a(5) from Giovanni Resta, Jun 11 2020

A054805 Second term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

37, 67, 97, 223, 277, 307, 457, 479, 613, 631, 719, 751, 853, 877, 929, 1087, 1297, 1423, 1447, 1471, 1543, 1657, 1663, 1693, 1733, 1777, 1783, 1847, 1861, 1867, 1987, 1993, 2053, 2137, 2333, 2371, 2377, 2459, 2467, 2503, 2521, 2531, 2579, 2609, 2647
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Comments

Second member of pairs of consecutive primes in A051634 (strong primes). - M. F. Hasler, Oct 27 2018

Links

Crossrefs

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Formula

a(n) = nextprime(A054804(n))= prevprime(A054806(n)), nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Extensions

Offset corrected to 1 by M. F. Hasler, Oct 27 2018
Definition clarified by N. J. A. Sloane, Aug 28 2021

A054807 Fourth term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

43, 73, 103, 229, 283, 313, 463, 491, 619, 643, 733, 761, 859, 883, 941, 1093, 1303, 1429, 1453, 1483, 1553, 1667, 1669, 1699, 1747, 1787, 1789, 1867, 1871, 1873, 1997, 1999, 2069, 2143, 2341, 2381, 2383, 2473, 2477, 2531, 2539, 2543, 2593, 2621, 2659
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Formula

a(n) = nextprime(A054806(n)), nextprime = A151800. - M. F. Hasler, Oct 27 2018

Extensions

Offset corrected to 1 by M. F. Hasler, Oct 27 2018
Definition clarified by N. J. A. Sloane, Aug 28 2021.

A054806 Third term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

41, 71, 101, 227, 281, 311, 461, 487, 617, 641, 727, 757, 857, 881, 937, 1091, 1301, 1427, 1451, 1481, 1549, 1663, 1667, 1697, 1741, 1783, 1787, 1861, 1867, 1871, 1993, 1997, 2063, 2141, 2339, 2377, 2381, 2467, 2473, 2521, 2531, 2539, 2591, 2617, 2657
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Programs

  • Mathematica
    Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,3]] (* Harvey P. Dale, Aug 28 2021 *)

Formula

a(n) = nextprime(A054805(n)) = prevprime(A054807(n)), nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Extensions

Offset corrected to 1 by M. F. Hasler, Oct 27 2018
Definition clarified by N. J. A. Sloane, Aug 28 2021

A054808 First term of strong prime quintets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Original entry on oeis.org

1637, 1759, 1831, 1847, 1979, 2357, 2447, 2477, 2503, 3413, 3433, 4177, 4493, 5237, 5399, 5419, 6011, 6619, 7219, 7253, 7727, 7853, 7907, 8123, 8467, 9551, 9587, 11003, 11353, 11551, 11813, 12379, 13841, 14797, 15107, 15511, 16007, 16273, 16787, 16993, 17359, 18149, 18289
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Comments

First member of pairs of consecutive primes in A054804 (first of strong quartets): The first 10^4 terms of that sequence yield over 2000 terms of this sequence. - M. F. Hasler, Oct 27 2018

Crossrefs

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

Programs

Formula

a(n) = prevprime(A054809(n)); A054808 = {m = A054804(n) | nextprime(m) = A054804(n+1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Extensions

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054835 Second term of weak prime septet: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3) < p(m+5)-p(m+4).

Original entry on oeis.org

15377, 64921, 68209, 68899, 128983, 128987, 143513, 154081, 158003, 192377, 221719, 222389, 244463, 249727, 285289, 318679, 337279, 354373, 357829, 374177, 385393, 394729, 402583, 402587, 419599, 439163, 441913, 448379, 457399, 457673, 458191, 482509, 527983, 529813, 577531, 582763, 655913
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051635, A229832.
Cf. A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Formula

a(1) = A229832(5). - Jonathan Sondow, Oct 13 2013
a(n) = A151800(A054834(n)) = A151799(A054836(n)), A151800 = nextprime, A151799 = prevprime; A054835 = { m = A054828(n) | m = nextprime(A054828(n-1)) }. - M. F. Hasler, Oct 27 2018

Extensions

More terms from M. F. Hasler, Oct 27 2018

A054838 Fifth term of weak prime septet: p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1).

Original entry on oeis.org

15401, 64951, 68227, 68917, 129001, 129011, 143537, 154111, 158029, 192407, 221737, 222437, 244493, 249763, 285343, 318701, 337301, 354391, 357883, 374219, 385417, 394747, 402601, 402613, 419623, 439199, 441953, 448421, 457421, 457697, 458219, 482527, 528001
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Programs

  • Mathematica
    Select[Partition[Prime[Range[7000]],7,1],Min[Differences[#,2]]>0&][[All,5]] (* Harvey P. Dale, Oct 15 2016 *)

Formula

a(n) = A151800(A054837(n)) = A151799(A054839(n)), A151800 = nextprime, A151799 = prevprime; A054838 = { m = A054831(n) | m = nextprime(A054831(n-1)) }. - M. F. Hasler, Oct 27 2018

Extensions

More terms from Harvey P. Dale, Oct 15 2016

A054801 Second term of balanced prime quartets: p(m)-p(m-1) = p(m+1)-p(m) = p(m+2)-p(m+1).

Original entry on oeis.org

257, 1747, 3307, 5107, 5387, 6317, 6367, 12647, 13457, 14747, 15797, 15907, 17477, 18217, 19477, 23327, 26177, 30097, 30637, 53617, 56087, 62207, 63697, 71347, 74471, 75527, 76561, 77557, 78797, 80917, 82787, 83437, 84437, 89107, 89387
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A006562, A054800-A054840.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[9000]],4,1],Length[ Union[ Differences[#]]] == 1&]][[2]] (* Harvey P. Dale, Oct 22 2013 *)

A054809 Second term of strong prime 5-tuples: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

Original entry on oeis.org

1657, 1777, 1847, 1861, 1987, 2371, 2459, 2503, 2521, 3433, 3449, 4201, 4507, 5261, 5407, 5431, 6029, 6637, 7229, 7283, 7741, 7867, 7919, 8147, 8501, 9587, 9601, 11027, 11369, 11579, 11821, 12391, 13859, 14813, 15121, 15527, 16033, 16301, 16811, 17011, 17377
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Comments

Initial member of pairs of consecutive primes in A054805 (second of quadruples): The first 10^4 terms of that sequence yield over 2000 terms of this sequence. - M. F. Hasler, Oct 27 2018

Links

Crossrefs

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quadruples (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime 4-tuples, 5-tuples, 6-tuples; A054819 .. A054840: members of weak prime 4-tuples, ..., 7-tuples.

Programs

  • Mathematica
    spqQ[n_]:=Module[{difs=Differences[n]},difs[[1]]>difs[[2]]> difs[[3]]> difs[[4]]]; Transpose[Select[Partition[Prime[ Range[2000]],5,1], spqQ]][[2]] (* Harvey P. Dale, May 06 2012 *)

Formula

a(n) = nextprime(A054808(n)) = prevprime(A054810(n)), nextprime = A151800, prevprime = A151799; A054809 = {m = A054805(n) | nextprime(m) = A054805(n+1)}. - M. F. Hasler, Oct 27 2018

Extensions

Corrected by Harvey P. Dale, May 06 2012
Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018
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