cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.

Original entry on oeis.org

5, 17, 41, 617, 71, 311, 2267, 521, 1877, 461, 1721, 347, 1151, 1787, 3581, 2141, 6449, 1319, 21377, 1487, 12251, 4799, 881, 23057, 659, 19541, 12377, 2381, 38747, 10529, 37361, 8627, 9041, 33827, 5879, 80231, 15359, 45821, 36107, 14627, 37991, 36527, 87251, 70997
Offset: 1

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

Smallest distance (A052380) and also smallest possible increment of twin-distances is 6.
Primes may occur between p+2 and p+6n.
The prime a(n) determines a prime quadruple: [p, p+2, p+6n, p+6n+2] and a [2, 6n-2, 2] d-pattern.

Examples

			The first 3 terms (5, 17, 41) are followed by difference patterns as it is displayed: 5 by [2, 4, 2], 17 by [2, 4+6, 2], 41 by [2, 4+6+6, 2] determining prime quadruples: (5, 7, 11, 13), (17, 19, 29, 31) or (41, 43, 59, 61), respectively.
a(10) = 461 gives the quadruple [461, 463, 521 = 461+60, 523], and between 521 and 463, 7 primes occur.

Links

Crossrefs

Cf. A001359, A007530, A052380, A052381, A053319, A113274, A113275.
Cf. A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k += 6]; k];p = 5; t = Table[0, {50}]; Do[ q = NextLowerTwinPrim[p]; d = (q - p)/6; If[d < 51 && t[[d]] == 0, t[[d]] = p; Print[{d, p}]]; p = q, {n, 1500}]; t (* Robert G. Wilson v, Oct 28 2005 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 5, q1 = 0, q2, d); forprime(p2 = 7, , if(p2 == p1 + 2, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Extensions

Name corrected by Amiram Eldar, Mar 04 2025

A052354 Least prime in A031928 (lesser of 10-twins) whose distance to the next 10-twin is 6*n.

Original entry on oeis.org

409, 691, 787, 547, 2053, 139, 4861, 283, 181, 25087, 337, 709, 2917, 829, 14197, 919, 3001, 33589, 2767, 421, 8221, 1879, 5179, 1249, 1471, 10141, 5011, 20533, 4483, 54091, 13249, 4663, 27883, 5869, 41443, 8599, 23311, 9049, 40699, 82591, 3109, 5323, 44917, 11971
Offset: 2

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

a(n) = p determines a prime quadruple [p, p+10, p+6n, p+6n+10] with difference pattern [10, 6n-10, 10].
The smallest distance between 10-twins [A052380(5)] is 12, while its increment is 6.
a(n) = p is the smallest of A031928 followed by the next 10-twin after a 6n distance.

Examples

			a(3) = 691 results in [691, 701, 709, 719] quadruple and [10, 8, 10] difference pattern without primes in the median gap.
a(11) = 25087 yields [25087, 25097, 25153, 25163] and [10, 56, 10] with 5 primes in the middle gap.

Links

Crossrefs

Cf. A031928, A052380, A052381, A053323.
Cf. A052350, A052351, A052352, A052353, A052355, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 10] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 1; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 05 2025 *)~
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 10, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 1; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Extensions

Name and offset corrected by Amiram Eldar, Mar 05 2025

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

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.

Examples

			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].

Links

Crossrefs

Cf. A031936, A052380, A052381, A053327.
Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052359.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

a(21) corrected and missing terms inserted by Sean A. Irvine, Nov 07 2021
Name and offset corrected by Amiram Eldar, Mar 05 2025

A052359 Least prime in A031938 (lesser of primes differing by 20) whose distance to the next 20-twin is 6*n.

Original entry on oeis.org

46703, 37223, 65147, 20369, 63929, 71999, 11597, 11027, 99767, 93503, 5903, 14087, 115163, 24821, 104891, 24923, 11867, 53381, 65657, 93581, 99623, 11447, 18461, 126761, 32213, 27653, 72797, 5717, 154247, 54449, 27827, 10223, 56747, 18617, 13421, 10433, 8543, 60107
Offset: 4

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

The smallest distance between 20-twins is 24 [= A052380(10)], while its minimal increment is 6.
a(n) = p starts [p, p+20, p+6n, p+6n+20] and [20, 6n-20, 20] patterns of primes and their difference.
a(n) = p is the smallest prime which starts a [p, p+20] twin followed by the next [p+6n, p+6n+20] twin.

Examples

			For n = 4, a(4) = 46703 results in prime quadruple [46703, 46723, 46727, 46747] and difference pattern [20, 4, 20].
For n = 14, a(14) = 5903 yields prime quadruple [5903, 5923, 5987, 6007] with 4 primes in the medial gap, and difference pattern [20, 64, 20].

Links

Crossrefs

Cf. A031938, A052380, A052381, A053331.
Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 20] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 3; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[15000] (* Amiram Eldar, Mar 05 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 20, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 3; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Extensions

Offset changed to 1 by Michel Marcus, Apr 30 2019
Name and offset corrected by Amiram Eldar, Mar 05 2025

A052351 Least prime in A023200 (lesser of 4-twins) such that the distance to the next 4-twin is 6*n.

Original entry on oeis.org

7, 67, 19, 43, 163, 127, 397, 229, 769, 1489, 673, 9547, 1009, 1783, 1693, 2857, 11677, 23869, 499, 1093, 4003, 28657, 10459, 29383, 12487, 6043, 41647, 7039, 17029, 19207, 15073, 24247, 65839, 29629, 18583, 9883, 66697, 100699, 7243, 53923, 82237, 6217, 76249
Offset: 1

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

a(n) is a "lesser of a 4-twin" prime whose distance to the next twin is 6n.
Both the smallest distance (A052380) and its increment for 4-twins is 6.
The prime a(n)=p is the first which determines a prime quadruple [p, p+4, p+6n, p+6n+4] and difference pattern of [4, 6n-4, 4].

Examples

			a(1) = 7 gives [7, 11,7+6 = 13, 17] with no primes between 11 and 13.
a(5) = 163 specifies [163, 167, 163+30 = 191, 193] with 4 primes between 167 and 193.

Links

Crossrefs

Cf. A023200, A052380, A052381, A053320.
Cf. A052350, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 4] // Flatten; pp = p[[i]]; dd = Differences[pp]/6; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 04 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 7, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 4, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Extensions

Name corrected by Amiram Eldar, Mar 04 2025

A052353 Least prime in A031926 (lesser of 8-twins) whose distance to the next 8-twin is 6*n.

Original entry on oeis.org

389, 683, 719, 359, 1523, 2699, 401, 929, 2153, 1373, 2459, 2531, 1439, 1733, 8573, 2741, 4943, 9059, 5051, 983, 3491, 9173, 7529, 761, 1823, 1571, 3041, 5399, 1193, 2273, 491, 8171, 23549, 5189, 5813, 53189, 3221, 4349, 32789, 49823, 18749, 19001, 10979, 89, 19433
Offset: 2

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

The smallest distance [A052380(4)] between 8-twins is 12, while its minimal increment is 6.
a(n) = p yields a prime quadruple of [p, p+8, p+6n, p+6n+8] and difference pattern of [8, 6n-8, 8].

Examples

			a(2) = 389 specifies quadruple of [389, 397, 401, 409] with no prime between 397 and 401;
a(11) = 1373 gives quadruple of [1373, 1381, 1439, 1447] and [8, 58, 8] difference pattern with 6 primes in the central gap.

Links

Crossrefs

Cf. A031926, A052380, A052381, A053322.
Cf. A052350, A052351, A052352, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 8] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 1; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 05 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 8, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 1; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Extensions

Name and offset corrected by Amiram Eldar, Mar 05 2025

A052355 Least prime in A031930 (lesser of 12-twins) whose distance to the next 12-twin is 2*n.

Original entry on oeis.org

199, 7937, 3331, 3049, 1511, 1789, 28607, 7001, 20599, 2069, 18257, 46477, 1201, 15569, 1459, 467, 23087, 23041, 2399, 6101, 7057, 6607, 23801, 3931, 3499, 9029, 5197, 7841, 3191, 1237, 3259, 45767, 4801, 1811, 1709, 40867, 23497, 125441, 5419, 3989, 18077, 21787
Offset: 6

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

The smallest distance between 12-twins [A052380(6)] is 12 and its minimal increment is 2.
a(n) = p specifies a quadruple [p, p+12, p+2n, p+2n+12] with difference pattern of [12, 2n-12, 12].

Examples

			a(7) = 7937 results in [7937, 7949, 7951, 7963] quadruple and [12, 2, 12] difference pattern.
a(10) = 1511 specifies [1511, 1523, 1531, 1543] quadruple and [12, 8, 12] difference pattern without prime in the central gap.

Links

Crossrefs

Cf. A031930, A052380, A052381, A053324.
Cf. A052350, A052351, A052352, A052353, A052354, A052356, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 12] // Flatten; pp = p[[i]]; dd = Differences[pp]/2 - 5; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[1q000] (* Amiram Eldar, Mar 05 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 12, q2 = p1; if(q1 > 0, d = (q2 - q1)/2 - 5; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Extensions

Name and offset corrected by Amiram Eldar, Mar 05 2025

A052356 Least prime in A031932 (lesser of 14-twins) whose distance to the next 14-twin is 6*n.

Original entry on oeis.org

24749, 293, 3833, 21467, 23417, 14159, 3779, 18353, 773, 4817, 18959, 2939, 863, 7607, 3677, 8039, 5939, 2633, 7727, 13367, 51839, 51659, 7043, 5153, 8447, 26189, 1409, 113, 7853, 1847, 13859, 43223, 2423, 24533, 65867, 50909, 19763, 15173, 15527, 86477, 55229
Offset: 3

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

The smallest distance between 14-twins [A052380(7)] is 18 and its minimal increment is 6.
a(n) = p is the first prime initiating [p, p+14, p+6n, p+6n+14] quadruple and prime difference pattern of [14, 6n-14, 14].

Examples

			n = 4 results in [293,307,317,331] primes pattern and [14,24,14] difference pattern with 2 further primes (311 and 313) in the central gap.

Links

Crossrefs

Cf. A031932, A052380, A052381, A053325.
Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052357, A052358, A052359.

Programs

  • Mathematica
    seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 14] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 2; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 05 2025 *)
  • PARI
    list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 14, 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

Extensions

Name and offset corrected by Amiram Eldar, Mar 05 2025

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Mar 07 2000

Keywords

Comments

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].

Examples

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

Links

Crossrefs

Cf. A031934, A052380, A052381, A053326.
Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052358, A052359.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

Incorrect 43207 removed and more terms from Sean A. Irvine, Nov 06 2021
Name and offset corrected by Amiram Eldar, Mar 05 2025
Showing 1-9 of 9 results.

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