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.

Showing 1-5 of 5 results.

A134122 Primes p such that q-p = 46, where q is the next prime after p.

Original entry on oeis.org

81463, 91033, 93001, 93637, 101221, 103723, 113233, 116593, 118297, 124021, 128053, 128767, 136897, 156841, 163063, 163927, 184777, 193891, 195817, 196201, 207877, 212923, 227743, 237091, 263323, 263443, 263677, 268297, 286927, 298513
Offset: 1

Views

Author

Rick L. Shepherd, Oct 08 2007

Keywords

Links

Crossrefs

Cf. A134121, A134123.

Programs

A134124 Primes p such that q-p = 50, where q is the next prime after p.

Original entry on oeis.org

31907, 45893, 60539, 69263, 95651, 112691, 162293, 167543, 172883, 181553, 197837, 206699, 212507, 220613, 225167, 246839, 272813, 274529, 291569, 293021, 298943, 334793, 338609, 345329, 349241, 355211, 362801, 368957, 369419, 380657
Offset: 1

Views

Author

Rick L. Shepherd, Oct 08 2007

Keywords

Links

Crossrefs

Cf. A134123.

Programs

  • Maple
    a:=proc(n) if nextprime(ithprime(n))-ithprime(n)=50 then ithprime(n) else end if end proc: seq(a(n),n=1..40000); # Emeric Deutsch, Oct 24 2007
  • Mathematica
    Transpose[Select[Partition[Prime[Range[40000]],2,1],#[[2]]-#[[1]] == 50&]][[1]] (* Harvey P. Dale, Feb 27 2015 *)
  • PARI
    is(n)=nextprime(n+1)==n+50 && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Sep 14 2015

A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 17, 19, 31, 89, 29, 37, 47, 359, 139, 41, 43, 53, 389, 181, 199, 59, 67, 61, 401, 241, 211, 113, 71, 79, 73, 449, 283, 467, 293, 1831, 101, 97, 83, 479, 337, 509, 317, 1933, 523, 107, 103, 131, 491, 409, 619, 773, 2113, 1069, 887
Offset: 1

Views

Author

Clark Kimberling, Mar 16 2010

Keywords

Comments

Every odd prime p = prime(i), i > 1, occurs in this array, in row (prime(i+1) - prime(i))/2. Polignac's conjecture states that each row contains an infinite number of indices. In case this does not hold, we can use the convention to continue finite rows with 0's, to ensure the sequence is well defined. - M. F. Hasler, Oct 19 2018
A permutation of the odd primes (A065091). - Robert G. Wilson v, Sep 13 2022

Examples

			Upper left hand corner of the array:
     3     5    11    17    29    41    59    71   101 ...
     7    13    19    37    43    67    79    97   103 ...
    23    31    47    53    61    73    83   131   151 ...
    89   359   389   401   449   479   491   683   701 ...
   139   181   241   283   337   409   421   547   577 ...
   199   211   467   509   619   661   797   997  1201 ...
   113   293   317   773   839   863   953  1409  1583 ...
  1831  1933  2113  2221  2251  2593  2803  3121  3373 ...
   523  1069  1259  1381  1759  1913  2161  2503  2861 ...
  (...)
Row 1: p(2) = 3, p(3) = 5, p(5) = 11, p(7) = 17,... these being the primes for which the next prime is 2 greater: (lesser of) twin primes A001359.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19,... these being the primes for which the next prime is 4 greater: (lesser of) cousin primes A029710.

Links

Crossrefs

Cf. A000040, A001223, A065091, A174349.
Rows 1, 2, 3, ...: A001359, A029710, A031924, A031926, A031928 (row 5), A031930, A031932, A031934, A031936, A031938 (row 10), A061779, A098974, A124594, A124595, A124596 (row 15), A126784, A134116, A134117, A134118, A126721 (row 20), A134120, A134121, A134122, A134123, A134124 (row 25), A204665, A204666, A204667, A204668, A126771 (row 30), A204669, A204670.
Rows 35, 40, 45, 50, ...: A204792, A126722, A204764, A050434 (row 50), A204801, A204672, A204802, A204803, A126724 (row 75), A184984, A204805, A204673, A204806, A204807 (row 100); A224472 (row 150).
Column 1: A000230.
Column 2: A046789.

Programs

  • Mathematica
    rows = 10; t2 = {}; Do[t = {}; p = Prime[2]; While[Length[t] < rows - off + 1, nextP = NextPrime[p]; If[nextP - p == 2*off, AppendTo[t, p]]; p = nextP]; AppendTo[t2, t], {off, rows}]; Table[t2[[b, a - b + 1]], {a, rows}, {b, a}] (* T. D. Noe, Feb 11 2014 *)
t[r_, 0] = 2; t[r_, c_] := Block[{p = NextPrime@ t[r, c - 1], q}, q = NextPrime@ p; While[ p + 2r != q, p = q; q = NextPrime@ q]; p]; Table[ t[r - c + 1, c], {r, 10}, {c, r, 1, -1}] (* Robert G. Wilson v, Nov 06 2020 *)
  • PARI
    A174350_row(g, N=50, i=0, p=prime(i+1), L=[])={g*=2; forprime(q=1+p, , i++; if(p+g==p=q, L=concat(L, q-g); N--||return(L)))} \\ Returns the first N terms of row g. - M. F. Hasler, Oct 19 2018

Formula

a(n) = A000040(A174349(n)). - Michel Marcus, Mar 30 2016

Extensions

Definition corrected and other edits by M. F. Hasler, Oct 19 2018

A329578 First of three consecutive primes with common gap 48.

Original entry on oeis.org

3565931, 3653863, 3985903, 5425613, 5647361, 6126971, 6292081, 6532553, 7133983, 7360363, 7389493, 7700131, 7865833, 7956163, 8467903, 8708291, 8972701, 9203743, 9603361, 9863551, 10279813, 10971743, 11998391, 12225251, 12474251, 12620843, 12966881, 13288211, 13376261, 13543451
Offset: 1

Views

Author

M. F. Hasler, Jan 02 2020

Keywords

Links

Crossrefs

Subsequence of A134123 (first of two primes with common gap 48).
A067388 (first of four primes with common gap 48) is a subsequence.
Cf. A047948, A052188, A052189, A052190, A052195, A052197, A052198, A089234 (analog for gaps 2, 4, 6, 12, 18, 24, ..., 60).

Programs

  • Magma
    [p:p in PrimesUpTo(14000000)| NextPrime(p)-p eq 48 and NextPrime(p+48)-p eq 96]; // Marius A. Burtea, Jan 03 2020
  • Mathematica
    Select[Partition[Prime[Range[900000]],3,1],Differences[#]=={48,48}&] [[All,1]] (* Harvey P. Dale, Aug 23 2021 *)
  • PARI
    vecextract( A134123, select(t->t==48, A134123[^1]-A134123[^-1], 1)) \\ Terms of A134123 with indices corresponding to first differences of 48: gives a(1..56) from A134123(1..10^4).

A273356 Numbers n such that n - 49, n - 1, n + 1, n + 49 are consecutive primes.

Original entry on oeis.org

913638, 2763882, 4500492, 6220518, 6473148, 13884468, 15131982, 15729942, 19671930, 20494602, 21372888, 23791350, 25541028, 29535348, 30787788, 30906768, 32085372, 34128168, 34139802, 34550430, 35989980, 37473180, 37784310, 38106372
Offset: 1

Views

Author

Karl V. Keller, Jr., May 20 2016

Keywords

Comments

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 belong to A249674 (divisible by 30).
The terms ending in 2 (resp. 8) are congruent to 12 (resp. 18) mod 30.
The numbers n - 49 and n + 1 belong to A134123 (p such that p + 48 is the next prime).
The numbers n - 49 and n - 1 belong to A062284 (p and p + 50 are primes).

Examples

			913638 is the average of the four consecutive primes 913589, 913637, 913639, 913687.
2763882 is the average of the four consecutive primes 2763833, 2763881, 2763883, 2763931.

Links

Crossrefs

Cf. A014574, A077800 (twin primes), A249674, A256753.

Programs

  • Mathematica
    Mean/@Select[Partition[Prime[Range[2325200]],4,1],Differences[#]=={48,2,48}&] (* Harvey P. Dale, Feb 10 2024 *)
  • PARI
    is(n)=isprime(n-1) && isprime(n+1) && precprime(n-2)==n-49 && nextprime(n+2)==n+49 \\ Charles R Greathouse IV, Jun 08 2016
  • Python
    from sympy import isprime,prevprime,nextprime
for i in range(0,60000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-49 and nextprime(i+1) == i+49: print (i,end=', ')
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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