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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A054810 Third term of strong prime 5-tuples: 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

1663, 1783, 1861, 1867, 1993, 2377, 2467, 2521, 2531, 3449, 3457, 4211, 4513, 5273, 5413, 5437, 6037, 6653, 7237, 7297, 7753, 7873, 7927, 8161, 8513, 9601, 9613, 11047, 11383, 11587, 11827, 12401, 13873, 14821, 15131, 15541, 16057, 16319
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

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[{a_,b_,c_,d_,e_}]:=(b-a)>(c-b)>(d-c)>(e-d); Transpose[ Select[ Partition[ Prime[ Range[2000]],5,1],spqQ]][[3]] (* Harvey P. Dale, Feb 25 2013 *)

Extensions

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

A054828 First term of weak prime sextet: 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

2903, 13463, 13901, 14947, 15373, 15377, 21397, 21557, 21859, 28277, 30869, 33199, 35591, 37691, 42221, 42569, 45821, 55661, 64661, 64919, 64921, 68207, 68209, 68897, 68899, 73939, 74201, 78577, 83089, 85513, 87313, 88001, 90907
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051635, A054800-A054840.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[9000]],6,1],AllTrue[ Differences[ #,2], Positive]&]] [[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 12 2014 *)

A054832 Fifth term of weak prime sextet: 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).

Original entry on oeis.org

2939, 13499, 13921, 14983, 15401, 15413, 21433, 21577, 21893, 28297, 30911, 33247, 35617, 37747, 42257, 42611, 45841, 55681, 64693, 64951, 64969, 68227, 68239, 68917, 68927, 73973, 74231, 78623, 83137, 85549, 87359, 88037, 90947
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. A051635, A054800-A054840.

Programs

  • Mathematica
    Select[Partition[Prime[Range[8800]],6,1],Min[Differences[#,2]]>0&][[All,5]] (* Harvey P. Dale, Feb 22 2020 *)

A054836 Third term of weak prime septet: p(m-1)-p(m-2) < 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).

Original entry on oeis.org

15383, 64927, 68213, 68903, 128987, 128993, 143519, 154087, 158009, 192383, 221723, 222403, 244471, 249737, 285301, 318683, 337283, 354377, 357839, 374189, 385397, 394733, 402587, 402593, 419603, 439171, 441923, 448387, 457403, 457679, 458197, 482513, 527987, 529819, 577537, 582767
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Links

Crossrefs

Cf. 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) = A151800(A054835(n)) = A151799(A054838(n)), A151800 = nextprime, A151799 = prevprime; A054836 = { m = A054829(n) | m = nextprime(A054829(n-1)) }. - M. F. Hasler, Oct 27 2018

Extensions

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

A054839 Sixth term of weak prime septet: p(m-4)-p(m-5) < 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).

Original entry on oeis.org

15413, 64969, 68239, 68927, 129011, 129023, 143551, 154127, 158047, 192431, 221747, 222461, 244507, 249779, 285377, 318713, 337313, 354401, 357913, 374239, 385433, 394759, 402613, 402631, 419651, 439217, 441971, 448451, 457433, 457711, 458239, 482539, 528013
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 quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

Programs

  • Mathematica
    Transpose[Select[Partition[Prime[Range[50000]],7,1],Min[ Differences[ #,2]]> 0&]][[6]] (* Harvey P. Dale, Sep 27 2015 *)

Formula

a(n) = A151800(A054838(n)) = A151799(A054840(n)), A054839 = { m = A054832(n) | m = A151800(A054832(n-1)) } (A151800: nextprime, A151799: prevprime). - M. F. Hasler, Oct 27 2018

Extensions

More terms from Harvey P. Dale, Sep 27 2015

A230852 Smallest k<3*2^n such that 3*2^n+k is the smallest of four consecutive primes in arithmetic progression or 0 if no solution.

Original entry on oeis.org

0, 0, 0, 0, 0, 59, 0, 0, 205, 229, 167, 353, 1595, 4459, 6407, 6215, 14995, 4559, 4697, 11399, 365, 10199, 19327, 39103, 3185, 13649, 15787, 2693, 21455, 24929, 32209, 30509, 13421, 5389, 36947, 12869, 27277, 38389, 973, 69199, 58835, 165629, 52597, 25463, 17923, 38629, 90263, 17153, 48143, 2171, 1255
Offset: 1

Views

Author

Pierre CAMI, Oct 31 2013

Keywords

Comments

Conjecture: if n>8 there is always a solution.

Examples

			3*2^6+59=251, and 251, 257, 263, 269 are four consecutive primes in arithmetic progression 6 so a(6)=59.
3*2^9+205=1741, and 1741, 1747, 1753, 1759 are four consecutive primes in arithmetic progression 6 so a(9)=205.

Links

Crossrefs

Cf. A054800, A230699.

Programs

  • PARI
    cpap4(p)=my(q=nextprime(p+1),g=q-p);nextprime(q+1)-q==g&&nextprime(p+2*g+1)==p+3*g
a(n)=forprime(p=3<Charles R Greathouse IV, Oct 31 2013

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).

A054811 Fourth term of strong prime quintets: p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m).

Original entry on oeis.org

1667, 1787, 1867, 1871, 1997, 2381, 2473, 2531, 2539, 3457, 3461, 4217, 4517, 5279, 5417, 5441, 6043, 6659, 7243, 7307, 7757, 7877, 7933, 8167, 8521, 9613, 9619, 11057, 11393, 11593, 11831, 12409, 13877, 14827, 15137, 15551, 16061, 16333
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Comments

First member of pairs of consecutive primes in A054807 (4th of strong prime quartets). - M. F. Hasler, Oct 27 2018

Links

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.

Formula

a(n) = nextprime(A054810(n)) = prevprime(A054812(n)), nextprime = A151800, prevprime = A151799; A054811 = {m = A054807(n) | nextprime(m) = A054807(n+1)}. - M. F. Hasler, Oct 27 2018

A054812 Fifth term of strong prime quintets: p(m-3)-p(m-4) > p(m-2)-p(m-3) > p(m-1)-p(m-2) > p(m)-p(m-1).

Original entry on oeis.org

1669, 1789, 1871, 1873, 1999, 2383, 2477, 2539, 2543, 3461, 3463, 4219, 4519, 5281, 5419, 5443, 6047, 6661, 7247, 7309, 7759, 7879, 7937, 8171, 8527, 9619, 9623, 11059, 11399, 11597, 11833, 12413, 13879, 14831, 15139, 15559, 16063, 16339
Offset: 1

Views

Author

Henry Bottomley, Apr 10 2000

Keywords

Comments

Second member of pairs of consecutive primes in A054807 (4th term of strong prime quartets). - M. F. Hasler, Oct 27 2018

Links

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

  • Mathematica
    spqQ[c_]:=Module[{d=Differences[c]},d[[1]]>d[[2]]>d[[3]]>d[[4]]]; Transpose[ Select[Partition[Prime[Range[2000]],5,1],spqQ]][[5]] (* Harvey P. Dale, Jan 01 2013 *)

Formula

a(n) = nextprime(A054811(n)); A054811 = {m = A054807(n) | prevprime(m) = A054807(n-1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018
Previous Showing 31-40 of 63 results. Next

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