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.

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

A335277 First index of strictly increasing prime quartets.

Original entry on oeis.org

7, 13, 22, 28, 49, 60, 64, 69, 70, 75, 78, 85, 89, 95, 104, 116, 122, 123, 144, 148, 152, 155, 173, 178, 182, 195, 201, 206, 212, 215, 219, 225, 226, 230, 236, 237, 244, 253, 256, 257, 265, 288, 302, 307, 315, 325, 328, 329, 332, 333, 336, 348, 355, 361, 373
Offset: 1

Views

Author

Gus Wiseman, May 30 2020

Keywords

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) < g(k + 1) < g(k + 2).

Examples

			The first 10 strictly increasing prime quartets:
   17  19  23  29
   41  43  47  53
   79  83  89  97
  107 109 113 127
  227 229 233 239
  281 283 293 307
  311 313 317 331
  347 349 353 359
  349 353 359 367
  379 383 389 397
For example, 107 is the 28th prime, and the primes (107,109,113,127) have differences (2,4,14), which are strictly increasing, so 28 is in the sequence.

Crossrefs

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly decreasing prime quartets are A335278.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Lengths of maximal weakly decreasing sequences of prime gaps are A333212.
Lengths of maximal strictly increasing sequences of prime gaps are A333253.
Cf. A000040, A006560, A031217, A054800, A054819, A059044, A084758, A089180.

Programs

  • Mathematica
    ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xPrimePi[x]]

Formula

prime(a(n)) = A054819(n).

A335278 First index of strictly decreasing prime quartets.

Original entry on oeis.org

11, 18, 24, 47, 58, 62, 87, 91, 111, 114, 127, 132, 146, 150, 157, 180, 210, 223, 228, 232, 242, 259, 260, 263, 269, 274, 275, 282, 283, 284, 299, 300, 309, 321, 344, 350, 351, 363, 364, 367, 368, 369, 375, 378, 382, 388, 393, 399, 406, 409, 413, 431, 442, 446
Offset: 1

Views

Author

Gus Wiseman, May 30 2020

Keywords

Comments

Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) > g(k + 1) > g(k + 2).

Examples

			The first 10 strictly decreasing prime quartets:
   31  37  41  43
   61  67  71  73
   89  97 101 103
  211 223 227 229
  271 277 281 283
  293 307 311 313
  449 457 461 463
  467 479 487 491
  607 613 617 619
  619 631 641 643
For example, 211 is the 47th prime, and the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 47 is in the sequence.

Crossrefs

Prime gaps are A001223.
Second prime gaps are A036263.
Strictly increasing prime quartets are A335277.
Equal prime quartets are A090832.
Weakly increasing prime quartets are A333383.
Weakly decreasing prime quartets are A333488.
Unequal prime quartets are A333490.
Partially unequal prime quartets are A333491.
Positions of adjacent equal prime gaps are A064113.
Positions of strict ascents in prime gaps are A258025.
Positions of strict descents in prime gaps are A258026.
Positions of adjacent unequal prime gaps are A333214.
Positions of weak ascents in prime gaps are A333230.
Positions of weak descents in prime gaps are A333231.
Indices of strictly decreasing rows of A066099 are A333256.
Lengths of maximal weakly increasing sequences of prime gaps are A333215.
Lengths of maximal strictly decreasing sequences of prime gaps are A333252.
Cf. A000040, A006560, A031217, A054800, A054804, A059044, A084758, A089180, A333253.

Programs

  • Mathematica
    ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>PrimePi[x]]

Formula

prime(a(n)) = A054804(n).

A293791 Prime 5-tuple 10000024493 + K * 30 for K = 0 to 4.

Original entry on oeis.org

10000024493, 10000024523, 10000024553, 10000024583, 10000024613
Offset: 1

Views

Author

Frank Ellermann, Oct 16 2017

Keywords

Comments

A052243(20) = 9843019 and A052243(21) = 9843049 are the first two primes in the smallest 5-tuple with difference 30 reported by Lander and Parkin in 1967. The much larger 5-tuple beginning with 10000024493 was reported by Jones, Lal and Blundon in the same year.
Sequence A059044 lists the quintuplets of consecutive primes in arithmetic progression (CPAP-5). A059044(9) ~ 10^8, A059044(86) ~ 10^9. a(1) ~ 10^10 might occur in that sequence around index n = 1000. - M. F. Hasler, Oct 28 2018

References

  • Yan S.Y. (2009) Number-Theoretic Preliminaries. In: Primality Testing and Integer Factorization in Public-Key Cryptography. Advances in Information Security, vol 11. Springer, Boston, MA.

Links

Crossrefs

Cf. A052243, A033290.

A366414 Primes p such that p and the four previous primes are in arithmetic progression.

Original entry on oeis.org

9843139, 37772549, 53868769, 71427877, 78364669, 79080697, 98150141, 99591553, 104437009, 106457629, 111267539, 121174931, 121174961, 168236239, 199450219, 203909011, 207068923, 216618307, 230952979, 234058991, 235524901, 253412437, 263651281, 268843153, 294485483, 296239907
Offset: 1

Views

Author

Harvey P. Dale, Oct 09 2023

Keywords

Examples

			9843019, 9843049, 9843079, 9843109, 9843139 are the 5 consecutive primes starting from A059044(1) and ending at a(1).

Crossrefs

Cf. A054643, A122535, A006562, A181424, A054803, A059044, A054800, A122535.

Programs

  • Mathematica
    Select[Partition[Prime[Range[10^7]],5,1],Length[Union[Differences[#]]]==1&][[;;,5]]
Previous Showing 11-16 of 16 results.

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