A031934 -id:A031934 - cp's OEIS frontend

A052357 Least prime in A031934 (lesser of 16-twins) whose distance to the next 16-twin is 6*n.

3373, 32917, 2221, 13597, 3391, 37783, 4057, 13537, 8581, 41911, 6763, 7333, 10867, 12457, 1831, 2113, 14683, 37201, 6637, 17581, 25423, 37447, 11353, 11197, 20611, 22453, 57397, 1933, 50707, 37591, 11503, 39733, 2593, 122131, 22921, 9013, 17167, 10273, 9661

Offset: 3

Labos Elemer, Mar 07 2000

nonn

The smallest distance between 16-twins [A052380(8)] is 18 and its minimal increment is 6.

a(n) = p is the smallest prime introducing the prime quadruple [p, p+16, p+6n, p+6n+16], which has a difference pattern [16, 6n-16, 16].

			a(9) = p = 4057 gives [4057, 4073, 4111, 4127] quadruple and [16, 38, 16] distance pattern with 4 primes in the medial gap.

Amiram Eldar, Table of n, a(n) for n = 3..1002

Cf. A031934, A052380, A052381, A053326.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052358, A052359.

seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 16] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 2; 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 + 16, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 2; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Incorrect 43207 removed and more terms from Sean A. Irvine, Nov 06 2021

Name and offset corrected by Amiram Eldar, Mar 05 2025

A053326 First differences of A031934.

Original entry on oeis.org

102, 180, 108, 30, 342, 210, 318, 252, 18, 42, 210, 414, 54, 456, 54, 402, 258, 342, 258, 756, 126, 78, 42, 450, 84, 576, 588, 66, 366, 228, 420, 246, 366, 240, 354, 90, 240, 156, 150, 198, 510, 246, 96, 828, 156, 60, 36, 870, 180, 114, 54, 660, 600, 522, 330

Offset: 1

Labos Elemer, Mar 06 2000

nonn

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

Cf. A031934.

Cf. A053321, A053322, A053323, A053324, A053325, A053327, A053331.

With[{p = Prime[Range[2000]]}, Differences[p[[Position[Differences[p], 16] // Flatten]]]] (* Amiram Eldar, Mar 10 2025 *)

A052260 Last filtering prime (A052180) of primes p such that next prime is p+16 (A031934).

Original entry on oeis.org

19, 29, 29, 23, 37, 23, 53, 53, 31, 43, 19, 41, 31, 23, 23, 41, 47, 67, 43, 71, 61, 67, 41, 83, 41, 41, 53, 31, 41, 71, 19, 19, 47, 43, 67, 83, 97, 23, 41, 37, 23, 19, 37, 29, 59, 29, 61, 23, 89, 37, 113, 89, 59, 71, 127, 71, 23, 89, 23, 73, 37, 19, 17, 73, 97, 137, 107, 37

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031934, A052180.

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

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

A204813 Primes followed by a gap of 256 = nextprime(p)-p.

Original entry on oeis.org

1872851947, 2362150363, 2394261637, 2880755131, 2891509333, 3353981623, 3512569873, 3727051753, 3847458487, 4008610423, 4486630573, 4541745583, 4755895531, 4837532347, 5227869607, 5389475977, 6201260587, 6229685347, 6952228483, 7325665111, 7414468513

Offset: 1

M. F. Hasler, Jan 19 2012

nonn

Washington Bomfim, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Cf. A204812, A204670, A126784, A031934, A031926, A029710, A001359; A062529, A138198, A123995.

list_gaps(g=256,f,N=25,p=0)=for(c=1,N,while(g+p!=p=nextprime(p+1),);if(f,write(f".txt",c" ",p-g),print1(", "p-g)))

a(8)-a(21) from Washington Bomfim

A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

Original entry on oeis.org

282, 295, 319, 331, 335, 378, 409, 445, 476, 478, 481, 510, 560, 566, 619, 624, 674, 701, 739, 775, 856, 871, 881, 886, 935, 941, 1007, 1069, 1077, 1121, 1146, 1193, 1222, 1261, 1286, 1322, 1331, 1356, 1372, 1388, 1405, 1460, 1487, 1500, 1587, 1603, 1608, 1612, 1699, 1719, 1734, 1740, 1811, 1876, 1924, 1956, 1969, 1977, 2002, 2022, 2034, 2042, 2071

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031934.

Vincenzo Librandi, Table of n, a(n) for n = 1..9790
Index entries for primes, gaps between.

Equals A000720 o A031934.
Row 8 of A174349.
Indices of 16'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..2100] | NthPrime(n+1) - NthPrime(n) eq 16]; // Vincenzo Librandi, Mar 19 2019

Select[Range[2500], Prime[#] + 16 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 19 2019 *)
Position[Partition[Prime[Range[2100]],2,1],?(#[[2]]-#[[1]]== 16&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Nov 01 2020 *)

A(N=100,g=16,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(A031934(n)).

A320706 = { i > 0 | prime(i+1) = prime(i) + 16 }.

A079018 Suppose p and q = p+16 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 17 possible difference patterns, namely [16], [4,12], [6,10], [10,6], [12,4], [4,2,10], [4,6,6], [4,8,4], [6,4,6], [6,6,4], [10,2,4], [4,2,4,6], [4,2,6,4], [4,6,2,4], [6,4,2,4], [4,2,4,2,4], [2,2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 7, 13, 31, 43, 67, 73, 151, 181, 211, 241, 277, 331, 463, 487, 1597, 1831

Offset: 1

Labos Elemer, Jan 24 2003

			p=181, q=197 has difference pattern [10,2,4] and {181,191,193,197} is the corresponding consecutive prime 4-tuple.

A022008(1)=7, A078952(1)=13, A078852(1)=73, A078953(1)=67, A078954(1)=1597, A078961(1)=31, A078856(1)=73, A078858(1)=151, A031934(1)=A000230(8)=1831.

Cf. A079016-A079024.

A231609 Table whose n-th row consists of primes p such that p + 2n is the next prime, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Nov 26 2013

The plot has an unusual gap near 10^5. Why?

			The following sequences are read by antidiagonals
{   3,    5,   11,   17,   29,   41,   59,   71,  101,  107, ...}
{   7,   13,   19,   37,   43,   67,   79,   97,  103,  109, ...}
{  23,   31,   47,   53,   61,   73,   83,  131,  151,  157, ...}
{  89,  359,  389,  401,  449,  479,  491,  683,  701,  719, ...}
{ 139,  181,  241,  283,  337,  409,  421,  547,  577,  631, ...}
{ 199,  211,  467,  509,  619,  661,  797,  997, 1201, 1237, ...}
{ 113,  293,  317,  773,  839,  863,  953, 1409, 1583, 1847, ...}
{1831, 1933, 2113, 2221, 2251, 2593, 2803, 3121, 3373, 3391, ...}
{ 523, 1069, 1259, 1381, 1759, 1913, 2161, 2503, 2861, 3803, ...}
{ 887, 1637, 3089, 3413, 3947, 5717, 5903, 5987, 6803, 7649, ...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A029710, A031924, A031926, A031928.
Cf. A031930, A031932, A031934, A031936, A031938.
Cf. A000230 (numbers in first column).

nn = 10; t = Table[{}, {nn}]; complete = 0; lastP = 3; While[complete < nn, p = NextPrime[lastP]; diff = p - lastP; If[diff <= 2*nn && Length[t[[diff/2]]] < nn - diff/2 + 1, AppendTo[t[[diff/2]], lastP]; If[Length[t[[diff/2]]] == nn - diff/2 + 1, complete++]]; lastP = p]; t2 = PadRight[t, {nn, nn}, 0]; Table[t2[[n-j+1, j]], {n, nn}, {j, n}]

cp's OEIS Frontend

A052357 Least prime in A031934 (lesser of 16-twins) whose distance to the next 16-twin is 6*n.

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A053326 First differences of A031934.

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A052260 Last filtering prime (A052180) of primes p such that next prime is p+16 (A031934).

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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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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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A204813 Primes followed by a gap of 256 = nextprime(p)-p.

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A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

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