A052380 -id:A052380 - cp's OEIS frontend

A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

3, 7, 47, 389, 409, 199, 24749, 3373, 20183, 46703, 19867, 16763, 142811, 14563, 69593, 763271, 276637, 255767, 363989, 383179, 247099, 2130809, 15370423, 3565931, 458069, 9401647, 6314393, 20823437, 9182389, 4911251, 15442121

Offset: 1

Labos Elemer, Mar 13 2000

nonn

A prime quadruple (triple), {[p,p+d],[p+D,p+D+d]} is called a "non-overlapping" (disjoint or touching) pair of twins if D = distance >= d = difference "inside" twin.

			If n=23, d=46, min{D}=48 then the first suitable quadruple of primes is [15370423, 15370469, 15370471, 15370517] with difference pattern [46, 2, 46]; if n=3, d=6, min{D}=6 then the first such triple is [47, 53, 53, 59] = [47, 53, 59] with difference pattern [6, 6].

The first 10 terms here appear as initial terms in A052350-A052359.

Smallest p so that [p, p+2n], [p+min{D}, p+2n+min{D}] is a quadruple (or triple if d=min{D}) of consecutive primes.

Corrected by Jud McCranie, Jan 04 2001

a(11) corrected by Sean A. Irvine, Nov 07 2021

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

Original entry on oeis.org

7, 13, 37, 97, 103, 223, 307, 457, 853, 877, 1087, 1297, 1423, 1483, 1867, 1993, 2683, 3457, 4513, 4783, 5227, 5647, 6823, 7873, 8287, 10453, 13687, 13873, 15727, 16057, 16063, 16183, 17383, 19417, 19423, 20743, 21013, 21313, 22273, 23053, 23557

Offset: 1

Labos Elemer, Mar 22 2000

nonn

The sequence includes A052166, A052168, A022008 and also other primes like 13, 103, 16063 etc.
a(n) is the lesser term of a 4-twin (A023200) after which the next 4-twin comes in minimal distance [here it is 2; see A052380(4/2)].
Analogous prime sequences are A047948, A052376, A052377 and A052188-A052198 with various [d, A052380(d/2), d] difference patterns following a(n).
All terms == 1 (mod 6) - Zak Seidov, Aug 27 2012
Subsequence of A022005. - R. J. Mathar, May 06 2017

			103 initiates [103,107,109,113] prime quadruple followed by [4,2,4] difference pattern.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A023200, A053320, A022008, A052166, A052168, A001223, A052380.

a = {}; Do[If[Prime[x + 3] - Prime[x] == 10, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Zerinvary Lajos, Apr 03 2007 *)
Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&][[All,1]] (* Harvey P. Dale, Jun 16 2017 *)

is(n)=n%6==1 && isprime(n+4) && isprime(n+6) && isprime(n+10) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) is the initial prime of a [p, p+4, p+6, p+6+4] prime-quadruple consisting of two 4-twins: [p, p+4] and [p+6, p+10].

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

Labos Elemer, Mar 07 2000

nonn

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.

			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.

Norman Luhn, Table of n, a(n) for n = 1..4624 (terms 1..4500 from Martin Raab).

Cf. A001359, A007530, A052380, A052381, A053319, A113274, A113275.

Cf. A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

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 *)

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

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

Labos Elemer, Mar 07 2000

nonn

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.

			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.

Amiram Eldar, Table of n, a(n) for n = 2..1001

Cf. A031928, A052380, A052381, A053323.

Cf. A052350, A052351, A052352, A052353, A052355, A052356, A052357, A052358, A052359.

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 *)~

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

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

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

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

Labos Elemer, Mar 07 2000

nonn

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.

			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].

Amiram Eldar, Table of n, a(n) for n = 4..1003

Cf. A031938, A052380, A052381, A053331.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358.

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 *)

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

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

Labos Elemer, Mar 07 2000

nonn

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].

			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.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A023200, A052380, A052381, A053320.

Cf. A052350, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

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 *)

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

Name corrected by Amiram Eldar, Mar 04 2025

A052352 Least prime in A031924 (lesser of 6-twins) such that the distance to the next 6-twin is 2*n.

Original entry on oeis.org

47, 23, 73, 61, 353, 31, 233, 131, 331, 653, 2441, 3733, 1033, 4871, 1063, 1621, 503, 607, 4211, 7823, 2287, 83, 383, 1231, 2903, 5981, 1123, 173, 11981, 11833, 1367, 2063, 4723, 19681, 2207, 2131, 2713, 9533, 6571, 1657, 23081, 15913, 7013, 14051, 9967, 22447

Offset: 3

Labos Elemer, Mar 07 2000

nonn

The increment of distance of 6-twins (A053321) is 2 (not 6), the smallest distance (A052380) is 6.
The middle gap 2n-6 may include primes, e.g., n = 12, a(12) = 653 and between 659 and 659 + 2*12 - 6 = 677, two primes occur (661 and 673).
a(n) = p yields a prime quadruple [p, p+6, p+2n, p+2n+6] with difference pattern [6, 2n-6, 6].

			For n = 3, 4, 5,  the quadruples are [47, 53, 53, 59] (a triple), [23, 29, 31, 37], [73, 79, 83, 89] with 53 - 47 = 6, 31 - 23 = 8 and 83 - 73 = 10 twin distances.

Amiram Eldar, Table of n, a(n) for n = 3..2000

Cf. A031924, A052380, A052381, A053321.

Cf. A052350, A052351, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 6] // Flatten; pp = p[[i]]; dd = Differences[pp]/2 - 2; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 04 2025 *)

list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 6, q2 = p1; if(q1 > 0, d = (q2 - q1)/2 - 2; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Name and offset 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

Labos Elemer, Mar 07 2000

nonn

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].

			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.

Amiram Eldar, Table of n, a(n) for n = 2..1001

Cf. A031926, A052380, A052381, A053322.

Cf. A052350, A052351, A052352, A052354, A052355, A052356, A052357, A052358, A052359.

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 *)

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

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

Labos Elemer, Mar 07 2000

nonn

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].

			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.

Amiram Eldar, Table of n, a(n) for n = 6..1005

Cf. A031930, A052380, A052381, A053324.

Cf. A052350, A052351, A052352, A052353, A052354, A052356, A052357, A052358, A052359.

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 *)

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

Name and offset corrected by Amiram Eldar, Mar 05 2025

cp's OEIS Frontend

A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

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A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

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

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

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

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

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

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