A052354 -id:A052354 - cp's OEIS frontend

A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.

6, 6, 6, 12, 12, 12, 18, 18, 18, 24, 24, 24, 30, 30, 30, 36, 36, 36, 42, 42, 42, 48, 48, 48, 54, 54, 54, 60, 60, 60, 66, 66, 66, 72, 72, 72, 78, 78, 78, 84, 84, 84, 90, 90, 90, 96, 96, 96, 102, 102, 102, 108, 108, 108, 114, 114, 114, 120, 120, 120, 126, 126, 126, 132

Offset: 1

Labos Elemer, Mar 13 2000

For d=D the quadruple of primes becomes a triple: [p,p+d],[p+d,p+2d].
Without the p > 3 condition, a(1)=2.
The starter prime p, is followed by a prime d-pattern of [d,D-d,d], where D-d=a(n)-2n is 4,2 or 0; these d-patterns are as follows: [2,4,2], [4,2,4], [6,6], [8,4,8], [10,2,10], [12,12], etc.
All terms of this sequence have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
a(n+1) is also the number of the circles added at the n-th iteration of the pattern generated by the construction rules: (i) At n = 0, there are six circles of radius s with centers at the vertices of a regular hexagon of side length s. (ii) At n > 0, draw a circle with center at each boundary intersection point of the figure of the previous iteration. The pattern seems to be the flower of life except at the central area. See illustration. - Kival Ngaokrajang, Oct 23 2015

			n=5, d=2n=10, the minimal distance for 10-twins is 12 (see A031928, d=10) the smallest term in A053323. It occurs first between twins of [409,419] and [421,431]; see 409 = A052354(1) = A052376(1) = A052381(5).

Kival Ngaokrajang, Illustration of initial terms
Sacred Geometry, Flower of life

Cf. A001223, A031924-A031938, A053319-A053331, A052350-A052358, A008620.

Table[2 n + 4 - 2 Mod[n + 2, 3], {n, 66}] (* Michael De Vlieger, Oct 23 2015 *)

vector(200, n, n--; 6*(n\3+1)) \\ Altug Alkan, Oct 23 2015

a(n) = 6*ceiling(n/3) = 6*ceiling(d/6) = D = D(n).
a(n) = 2n + 4 - 2((n+2) mod 3). - Wesley Ivan Hurt, Jun 30 2013
a(n) = 6*A008620(n-1). - Kival Ngaokrajang, Oct 23 2015

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

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

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

Labos Elemer, Mar 07 2000

nonn

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].

			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.

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

Cf. A031932, A052380, A052381, A053325.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052357, A052358, A052359.

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 *)

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

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

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

cp's OEIS Frontend

A052380 a(n) = D is the smallest distance (D) between 2 non-overlapping prime twins differing by d=2n; these twins are [p,p+d] or [p+D,p+D+d] and p > 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links