A126771 -id:A126771 - cp's OEIS frontend

A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

253444777, 271386581, 286000489, 415893013, 475992773, 523294549, 620164949, 794689481, 838188877, 840725323, 846389227, 884106599, 884951807, 908725507, 941796223, 952288331, 971614151, 1002290693, 1003166771, 1006976797, 1053792359, 1097338313, 1163141201

Offset: 1

Zak Seidov, May 09 2012

nonn

Subsequence of A089234 which itself is a subsequence of A126771:
a(1) = 253444777 = A089234(417) = A126771(81526),
a(36) = 1998782563 = A089234(5579) = A126771(788920).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A089234, A126771.
Analogous sequences (start of CPAP-4, with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54].
Cf. A054800: union of all sequences of this type (start of CPAP-4).

A210683(n, p=2, v=1, g=60, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A210683(1, p) to get the next term, e.g.:
p=0; A210683_vec=vector(10,i,p=A210683(1,p)) \\ Will take a long time! - M. F. Hasler, Oct 26 2018

A089234 Primes p such that p, p+60, p+120 are consecutive primes.

Original entry on oeis.org

4911251, 5309539, 9113263, 11355797, 11397103, 13940057, 14306203, 14313527, 14585089, 17172521, 21126109, 24419281, 24581803, 24861631, 24922291, 25308799, 26241751, 26722523, 27408193, 28740919, 29675137, 30045811

Offset: 1

Pierre CAMI, Dec 11 2003

nonn

Subsequence of A126771: a(1)=4911251=A126771(747),a(2)=5309539=A126771(824), a(3)=9113263=A126771(1618), etc. - Zak Seidov, May 07 2012

			4911251, 4911311, 4911371 are consecutive primes with 2 consecutive gaps of 60.

Zak Seidov and Pierre CAMI, Table of n, a(n) for n = 1..5000 (terms 1001 to 2899 from Pierre CAMI)
OEIS wiki, Consecutive primes in arithmetic progression, updated Jan. 2020

Cf. A126771.

A210727 Primes p such that p, p+60, p+120, p+180, p+240 are consecutive primes.

Original entry on oeis.org

6182296037, 6675135377, 6798668171, 10301484257, 12665852879, 14922537067, 26348961209, 27009595127, 30321479693, 35572512473, 36938181239, 37962662791, 45320751701, 45999570191, 50772316757, 52628649973, 55745449033, 56425976891, 57984707603, 60553081499

Offset: 1

Zak Seidov, May 10 2012

nonn

Subsequence of A210683: a(1) = 6182296037 = A210683(146), a(2) = 6675135377 = A210683(166), a(3) = 6798668171 = A210683(175).

The minimal possible value of the first differences of a set of six consecutive primes in arithmetic progression is 30 (see A058362 for examples). - Jon E. Schoenfield, Jan 04 2024

Zak Seidov, Table of n, a(n) for n = 1..300

Cf. A058362, A089234, A126771, A210683.

A204672 Primes followed by a gap of 120.

Original entry on oeis.org

1895359, 2898239, 6085441, 7160227, 7784039, 7803491, 7826899, 8367397, 8648557, 9452959, 10052071, 10863973, 11630503, 11962823, 12109697, 12230233, 12415681, 14411737, 14531899, 15014557, 15020737, 15611909, 16179041

Offset: 1

M. F. Hasler, Jan 18 2012

nonn

M. F. Hasler, Table of n, a(n) for n = 1..433 (all a(n) < 10^8).
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n).
Cf. A126771 (gap 60), A126724 (gap 150), A204673 (gap 180).
Cf. A001359, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

N = 2*10^7; % to get all terms <= N
P = primes(N+120);
J = find(P(2:end) - P(1:end-1) == 120);
P(J)  % Robert Israel, Feb 28 2017

Transpose[Select[Partition[Prime[Range[1100000]],2,1],Last[#]-First[#] == 120&]] [[1]] (* Harvey P. Dale, Jul 11 2014 *)

g=120;c=o=0;forprime(p=1,default(primelimit),(-o+o=p)==g&write("c:/temp/b204672.txt",c++" "p-g))

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

A204668 Primes p such that q-p = 58, where q is the next prime after p.

Original entry on oeis.org

44293, 58831, 79699, 85933, 153001, 181303, 190921, 214993, 221101, 237409, 277099, 302053, 307759, 311041, 399793, 402139, 413353, 420613, 422479, 442633, 492319, 510619, 525043, 531919, 558343, 600109, 624859, 640993, 650701, 677473, 686209, 696181, 714283, 724021, 733009, 737353

Offset: 1

N. J. A. Sloane, Jan 17 2012

nonn

M. F. Hasler, Table of n, a(n) for n = 1..14611 (all terms < 10^8).
Index entries for primes, gaps between

Cf. A126771 for gaps of 60.

g=58;c=o=0;forprime(p=1,default(primelimit),(-o+o=p)==g&write("c:/temp/b204668.txt",c++" "p-g))

A224472 Primes followed by a gap of 300.

Original entry on oeis.org

4758958741, 5612345261, 6169169561, 6306815239, 6646984159, 7335508261, 8645089003, 8806019249, 9047808247, 9148138313, 9466071347, 9907846261, 10055451683, 11063821453, 11475026363, 11603081459, 12292390637, 12750876857, 13833827471, 14636472007, 15876700949

Offset: 1

Zak Seidov, Apr 07 2013

nonn

The first twin gap equal to 300 occurs for p = 6537587646371. - Giovanni Resta, Apr 07 2013

Zak Seidov, Table of n, a(n) for n = 1..3718
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n), A126771 (gap 60), A126724 (gap 150), A204673 (gap 180), A204807 (gap 200), A000230, A001359, A204672, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

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A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

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A089234 Primes p such that p, p+60, p+120 are consecutive primes.

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A210727 Primes p such that p, p+60, p+120, p+180, p+240 are consecutive primes.

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A204672 Primes followed by a gap of 120.

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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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A204668 Primes p such that q-p = 58, where q is the next prime after p.

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A224472 Primes followed by a gap of 300.

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