A031936 -id:A031936 - cp's OEIS frontend

A052358 Least prime in A031936 (lesser of 18-twins) whose distance to the next 18-twin is 2*n.

20183, 20963, 14011, 26759, 7433, 45613, 4703, 21911, 26539, 18233, 6581, 4423, 7351, 37379, 55903, 25801, 4373, 6529, 35879, 119993, 22171, 12923, 10691, 52609, 14303, 20201, 16231, 21121, 103049, 17863, 6451, 34439, 50341, 76129, 3803, 23251, 15241, 14369

Offset: 9

Labos Elemer, Mar 07 2000

nonn

The smallest distance between 18-twins [A052380(9)] is 18 and its minimal increment is 2.

a(n) = p is the first prime initiating [p, p+18, p+2n, p+2n+18] prime and [18, 2n-18, 18] d-pattern.

			a(11) = 14011 initiates prime quadruple [14011, 14029, 14033, 14051] and difference pattern [18, 4, 18].
a(15) = 4703 specifies prime quadruple  [4703, 4721, 4133, 4151] which includes 2 primes (4723, 4729) in the center, and difference pattern [18, 28, 18].

Amiram Eldar, Table of n, a(n) for n = 9..1008

Cf. A031936, A052380, A052381, A053327.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052359.

seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 18] // Flatten; pp = p[[i]]; dd = Differences[pp]/2 - 8; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[12000] (* Amiram Eldar, Mar 05 2025 *)

list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 18, q2 = p1; if(q1 > 0, d = (q2 - q1)/2 - 8; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

a(21) corrected and missing terms inserted by Sean A. Irvine, Nov 07 2021

Name and offset corrected by Amiram Eldar, Mar 05 2025

A053327 First differences of A031936.

Original entry on oeis.org

546, 190, 122, 378, 154, 248, 342, 358, 942, 86, 270, 214, 50, 40, 140, 100, 30, 326, 150, 274, 528, 218, 222, 78, 52, 38, 540, 192, 42, 40, 26, 162, 262, 308, 570, 348, 184, 456, 200, 244, 498, 62, 378, 1488, 52, 50, 42, 160, 60, 780, 78, 42, 128, 22, 270, 66

Offset: 1

Labos Elemer, Mar 06 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A031936.

Cf. A053321, A053322, A053323, A053324, A053325, A053326, A053331.

With[{p = Prime[Range[2000]]}, Differences[p[[Position[Differences[p], 18] // Flatten]]]] (* Amiram Eldar, Mar 10 2025 *)

A153418 Primes p such that p+18 is also prime.

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 41, 43, 53, 61, 71, 79, 83, 89, 109, 113, 131, 139, 149, 163, 173, 179, 181, 193, 211, 223, 233, 239, 251, 263, 293, 313, 331, 349, 379, 383, 401, 421, 431, 439, 443, 449, 461, 491, 503, 523, 569, 599, 601, 613, 641, 643, 659, 673, 683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Both p and p+18 have the same digital root (A010888). - Zak Seidov, Sep 14 2015
No term belongs to A030432. - Michel Marcus, Sep 14 2015
No term belongs to A045437. - Bruno Berselli, Sep 14 2015

			5 is in sequence because 5+18=23 is also prime;
11 is in sequence because 11+18=29 is also prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007953, A010888, A045437, A049488, A030432, A153417.

A031936 is a subsequence. - Zak Seidov, Sep 13 2015

[p: p in PrimesUpTo(1000) | IsPrime(p+18)]; // Vincenzo Librandi, Apr 14 2013

lst={};d=18;Do[p=Prime[n];If[PrimeQ[p+d],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[150]], PrimeQ[(# + 18)]&] (* Vincenzo Librandi, Apr 14 2013 *)

list(n)=forprime(p=1,n,if(isprime(p+18),print1(p", ")))  \\ Anders Hellström, Sep 13 2015

[p for p in primes(700) if is_prime(p+18)]; # Bruno Berselli, Sep 14 2015

Definition improved by Bruno Berselli, Oct 31 2012

A052189 Primes p such that p, p+18, p+36 are consecutive primes.

Original entry on oeis.org

20183, 21893, 25373, 29251, 30431, 34613, 50423, 54833, 56131, 58111, 63541, 66413, 74453, 74471, 76543, 76561, 77933, 78241, 81421, 107563, 108421, 110441, 112163, 121403, 122081, 122561, 131023, 132893, 132911, 135283, 137303, 137831, 143141, 144593, 145643

Offset: 1

Labos Elemer, Jan 28 2000

nonn

Old name was "Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=18."

			20183 is a term since , 20183, 20201, and 20219 are consecutive primes with difference of 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A031936
A033448 is a subsequence.
Cf. A001223, A033451, A047948, A052188.

Select[Partition[Prime[Range[15000]], 3, 1], Differences[#] == {18, 18} &][[;; , 1]] (* Amiram Eldar, Feb 28 2025 *)

list(lim) = {my(p1 = 2, p2 = 3); forprime(p3 = 5, lim, if(p2 - p1 == 18 && p3 - p2 == 18, print1(p1, ", ")); p1 = p2; p2 = p3);} \\ Amiram Eldar, Feb 28 2025

Name changed by Jon E. Schoenfield, May 30 2018

A098974 Primes p such that q-p = 24, where q is the next prime after p.

Original entry on oeis.org

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

A126784 Primes p such that q-p = 32, where q is the next prime after p.

Original entry on oeis.org

5591, 10799, 27701, 27851, 33647, 39047, 41081, 41687, 43721, 44417, 45989, 47459, 50789, 52457, 55259, 55547, 61781, 62351, 64817, 66239, 67307, 69959, 73907, 79907, 80567, 82307, 84089, 88037, 94169, 94961, 99191, 99929, 100559, 102611

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Feb 24 2007

Lower prime of a difference of 32 between consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A029710, A031924, A031926, A031928, A031930, A031932, A031934, A031936, A031938, A061779, A098974, A124594, A124595, A124596, A000230, A050434.

lista(nn) = {p = 2; while (p < nn, q = nextprime(p+1); if (q - p == 32, print1(p, ", ")); p = q;);} \\ Michel Marcus, Jul 17 2013

A031937 Upper prime of a difference of 18 between consecutive primes.

Original entry on oeis.org

541, 1087, 1277, 1399, 1777, 1931, 2179, 2521, 2879, 3821, 3907, 4177, 4391, 4441, 4481, 4621, 4721, 4751, 5077, 5227, 5501, 6029, 6247, 6469, 6547, 6599, 6637, 7177, 7369, 7411, 7451, 7477, 7639, 7901, 8209, 8779, 9127, 9311, 9767

Offset: 0

Jeff Burch

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for primes, gaps between

Cf. A031935.

Transpose[Select[Partition[Prime[Range[1300]],2,1],Last[#]-First[#] == 18&]][[2]] (* Harvey P. Dale, Oct 06 2011 *)

a(n) = A031936(n)+18.

A117838 Smaller of two consecutive prime numbers with the same digital root.

Original entry on oeis.org

523, 1069, 1259, 1381, 1759, 1913, 2161, 2503, 2861, 3803, 3889, 4159, 4373, 4423, 4463, 4603, 4703, 4733, 5059, 5209, 5483, 6011, 6229, 6451, 6529, 6581, 6619, 7159, 7351, 7393, 7433, 7459, 7621, 7883, 8191, 8761, 9109, 9293, 9551, 9749, 9949

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 30 2006

Contains all sequences with primes that are followed by a prime gap which is a multiple of 18 - since adding multiples of 9 does not change the digital root and the gaps are even. So A031936 (gap 18) and A134117 (gap 36) are subsequences and lower primes of prime gap 54 (35617, 40289, 40639, 86869, 100853,...), prime gap 72 (31397, 360091, 507217, 517639, 633667, 650107, 705317....) or prime gap 90 (404851,576791,..), for example, are also in here (cf. A000230). - R. J. Mathar, Apr 14 2008

			523 and 541 are two consecutive prime numbers with the same digital root, namely 1.

M. F. Hasler, Table of n, a(n) for n = 1..5833.

Select[Prime[Range[1250]],Mod[ # - 1, 9] + 1 ==Mod[NextPrime[#]-1,9]+1&] (* James C. McMahon, Sep 14 2024 *)

isA117838(p)={ (nextprime(p+1)-p)%9==0 }
forprime( p=1,10^4, isA117838(p) & print1(p", ")) \\ M. F. Hasler, Apr 13 2008

{A000040(i): 18 | A001223(i), any i}. - R. J. Mathar, Apr 14 2008

Corrected by R. J. Mathar and M. F. Hasler, Apr 13 2008

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

A320707 Indices of primes followed by a gap (distance to next larger prime) of 18.

Original entry on oeis.org

99, 180, 205, 221, 274, 293, 326, 368, 416, 529, 539, 573, 597, 602, 607, 623, 635, 639, 677, 693, 725, 785, 811, 838, 844, 852, 855, 916, 937, 939, 942, 945, 968, 997, 1028, 1093, 1130, 1151, 1203, 1227, 1252, 1304, 1311, 1349, 1508, 1514, 1519, 1523, 1540, 1547, 1629, 1636, 1641, 1654, 1656

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031936.

Vincenzo Librandi, Table of n, a(n) for n = 1..10300
Index entries for primes, gaps between.

Equals A000720 o A031936.
Row 9 of A174349.
Indices of 18's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1700] | NthPrime(n+1) - NthPrime(n) eq 18]; // Vincenzo Librandi, Mar 22 2019

Select[Range[1700], Prime[#] + 18 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)
Flatten[Position[Differences[Prime[Range[2000]]],18]] (* Harvey P. Dale, May 12 2022 *)

A(N=100,g=18,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031936(n)).

A320707 = { i > 0 | prime(i+1) = prime(i) + 18 } = A001223^(-1)({18}).

cp's OEIS Frontend

A052358 Least prime in A031936 (lesser of 18-twins) whose distance to the next 18-twin is 2*n.

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A053327 First differences of A031936.

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A153418 Primes p such that p+18 is also prime.

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A052189 Primes p such that p, p+18, p+36 are consecutive primes.

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A098974 Primes p such that q-p = 24, where q is the next prime after p.

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A126784 Primes p such that q-p = 32, where q is the next prime after p.

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A031937 Upper prime of a difference of 18 between consecutive primes.

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A117838 Smaller of two consecutive prime numbers with the same digital root.

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