A134122 -id:A134122 - cp's OEIS frontend

A134121 Primes p such that q-p = 44, where q is the next prime after p.

15683, 36389, 37907, 76163, 92507, 108587, 109673, 124853, 138683, 138977, 140009, 140477, 183203, 186959, 203669, 225383, 226697, 229037, 232259, 242819, 242927, 244043, 245339, 271289, 273653, 275837, 279353, 280139, 282617, 285377

Offset: 1

Rick L. Shepherd, Oct 08 2007

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A134120, A134122.

is(n)=nextprime(n+1)==n+44 && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

A134123 Primes p such that q-p = 48, where q is the next prime after p.

Original entry on oeis.org

28229, 73189, 86629, 105769, 106543, 113843, 137029, 156371, 157579, 163259, 166099, 168803, 172441, 177043, 177691, 179849, 180569, 183713, 203713, 204251, 206651, 220973, 222199, 226943, 229849, 233021, 234383, 240209, 242009, 260269

Offset: 1

Rick L. Shepherd, Oct 08 2007

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A134122, A134124.

Transpose[Select[Partition[Prime[Range[25000]],2,1],Last[#]-First[#]==48&]][[1]]  (* Harvey P. Dale, Feb 07 2011 *)
Select[Prime[Range[25000]], NextPrime[#] - # == 48 &]

is(n)=nextprime(n+1)==n+48 && isprime(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

Clark Kimberling, Mar 16 2010

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

			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.

Robert G. Wilson v, Falling antidiagonals 1..115, flattened (first 50 from T. D. Noe).
Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.

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.

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 *)

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

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

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

A273355 Numbers n such that n - 47, n - 1, n + 1, n + 47 are consecutive primes.

Original entry on oeis.org

15370470, 15462870, 18216510, 23726160, 30637050, 31054740, 38907060, 39220080, 44499900, 44678190, 60563100, 66248550, 86219910, 87095190, 87948780, 93773970, 96802860, 103011990, 105953760, 105978330, 106960410, 111219990, 116281770

Offset: 1

Karl V. Keller, Jr., May 20 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs), A249674 (divisible by 30) and A256753.
The numbers n - 47 and n + 1 belong to A134122 (p such that p + 46 is the next prime).
The numbers n - 47 and n - 1 belong to primes p such that p and p + 48 are primes.

			15370470 is the average of the four consecutive primes 15370423, 15370469, 15370471, 15370517.
15462870 is the average of the four consecutive primes 15462823, 15462869, 15462871, 15462917.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

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

is(n)=isprime(n-1) && isprime(n+1) && precprime(n-2)==n-47 && nextprime(n+2)==n+47 \\ Charles R Greathouse IV, Jun 08 2016

from sympy import isprime,prevprime,nextprime
for i in range(0,160000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-47 and nextprime(i+1) == i+47: print (i,end=', ')

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A134121 Primes p such that q-p = 44, where q is the next prime after p.

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A134123 Primes p such that q-p = 48, where q is the next prime after p.

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

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A273355 Numbers n such that n - 47, n - 1, n + 1, n + 47 are consecutive primes.

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