A046789 -id:A046789 - cp's OEIS frontend

A047948 Smallest of three consecutive primes with a difference of 6: primes p such that p+6 and p+12 are the next two primes.

47, 151, 167, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4451, 4591, 4651, 4987, 5101, 5107, 5297, 5381, 5387, 5557, 5801, 6067, 6257, 6311, 6317

Offset: 1

Enoch Haga

Let p(k) be the k-th prime; sequence gives p(k) such that p(k+2) - p(k+1) = p(k+1) - p(k) = 6.

			47 is a term as the next two primes are 53 and 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A031924.
A033451 (four consecutive primes with difference 6) is a subsequence.
Cf. A078853, A046789.
Cf. A001223, A033451, A052197, A052198.

ok[p_] := (q = NextPrime[p]) == p+6 && NextPrime[q] == q+6; Select[Prime /@ Range[1000], ok][[;; 45]] (* Jean-François Alcover, Jul 11 2011 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={6,6}&]] [[1]] (* Harvey P. Dale, Apr 25 2014 *)

is_A047948(n)={nextprime(n+1)==n+6 && nextprime(n+7)==n+12 && isprime(n)} \\ Charles R Greathouse IV, Aug 17 2011, simplified by M. F. Hasler, Jan 13 2013

p=2;q=3;forprime(r=5,1e4,if(r-p==12&&q-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, Aug 17 2011

Corrected by T. D. Noe, Mar 07 2008

A046728 Let p1, p2 be first pair of consecutive primes with difference 2n; let p3, p4 be 2nd such pair; sequence gives "wadi" value p3-p1.

Original entry on oeis.org

2, 6, 8, 270, 42, 12, 180, 102, 546, 750, 822, 510, 660, 300, 534, 5208, 7140, 3302, 7800, 1476, 18840, 20706, 9570, 44960, 13986, 5862, 4672, 71118, 14538, 59370, 156348, 17688, 59184, 155736, 1782, 328694, 110370, 119616, 93402, 76116

Offset: 1

Enoch Haga

nonn

			First consecutive prime pair having difference 60 is p1=43331, p2=43391; next is p3=102701, p4=102761; so a(30)=102701-43331=59370.

Enoch Haga, Exploring Prime Numbers on Your PC, 2nd edition, 1998, ISBN 1-885794-16-9, Table 5, pp. 40-41.

Robert G. Wilson v, Table of n, a(n) for n = 1..201

Cf. A038460, A038343. a(i)=b(i)-c(i) where b(i) is A046789 and c(i) is A000230.

Flatten[Table[Differences[First /@ Take[Select[Partition[Prime[Range[60000]], 2, 1], Differences[#] == {2 n} &], 2]], {n, 40}]] (* Jayanta Basu, Jun 27 2013 *)

a(n)=my(k,p=2,p1);forprime(q=3,,if(q-p==2*n,if(k++>1,return(p-p1),p1=p));p=q) \\ Charles R Greathouse IV, Jun 27 2013

a(n) = A046789(n) - A000230(n). - Robert G. Wilson v, Nov 26 2020

More terms from Jud McCranie

a(31) inserted by Jayanta Basu, Jun 27 2013

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

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A047948 Smallest of three consecutive primes with a difference of 6: primes p such that p+6 and p+12 are the next two primes.

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A046728 Let p1, p2 be first pair of consecutive primes with difference 2n; let p3, p4 be 2nd such pair; sequence gives "wadi" value p3-p1.

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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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