A053323 -id:A053323 - cp's OEIS frontend

A031928 Lower prime of a difference of 10 between consecutive primes.

139, 181, 241, 283, 337, 409, 421, 547, 577, 631, 691, 709, 787, 811, 829, 919, 1021, 1039, 1051, 1153, 1171, 1249, 1399, 1471, 1627, 1699, 1723, 1801, 1879, 2017, 2029, 2053, 2089, 2143, 2521, 2647, 2719, 2731, 2767, 2887, 2917, 3001, 3109, 3361, 3517, 3547, 3571, 3583, 3709, 3769, 3823, 3853, 4201, 4219, 4231, 4243, 4261, 4273, 4327, 4339, 4363, 4483, 4663, 4861, 4909, 4957, 5011, 5179, 5323, 5581, 5659, 5701, 5791, 5869, 6079, 6091

Offset: 1

Lekraj Beedassy, Jul 23 2003

Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n). Namely, a(n)^(1/n) is a strictly decreasing function of n (see comments at A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A023203, A053323, A248855, A290450.

[p: p in PrimesUpTo(7000) | NextPrime(p)-p eq 10]; // Bruno Berselli, Apr 09 2013

Transpose[Select[Partition[Prime[Range[800]], 2, 1], #[[2]] - #[[1]] == 10&]] [[1]] (* Harvey P. Dale, Oct 02 2014 *)
p = Prime@Range@800; p[[Flatten@Position[Differences@p, 10]]] (* Hans Rudolf Widmer, Aug 28 2022 *)

forprime(p=o=1,1e4,10+o==(o=p)&&print1(p-10",")) \\ M. F. Hasler, Mar 10 2017

a(n) = prime(A320703(n)). - R. J. Mathar, Apr 30 2024

Edited by Labos Elemer, Jul 25 2003

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.

Original entry on oeis.org

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

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

A053325 First differences of A031932.

Original entry on oeis.org

180, 24, 456, 66, 24, 90, 456, 174, 264, 192, 318, 66, 210, 120, 66, 120, 84, 570, 84, 102, 54, 30, 276, 354, 324, 72, 84, 180, 156, 24, 336, 270, 114, 666, 324, 150, 90, 324, 96, 24, 126, 186, 108, 126, 24, 150, 162, 528, 186, 54, 120, 90, 300, 456, 120, 150

Offset: 1

Labos Elemer, Mar 06 2000

nonn

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

Cf. A031932.

Cf. A053321, A053322, A053323, A053324, A053326, A053327, A053331.

Differences[Transpose[Select[Partition[Prime[Range[1500]],2,1], Last[#]- First[#] == 14&]][[1]]] (* Harvey P. Dale, Aug 24 2012 *)

A053321 First differences of A031924.

Original entry on oeis.org

8, 16, 6, 8, 12, 10, 48, 20, 6, 10, 6, 60, 18, 6, 6, 8, 60, 22, 14, 6, 10, 50, 10, 60, 38, 16, 6, 8, 16, 6, 8, 6, 40, 6, 24, 50, 6, 18, 190, 6, 24, 6, 14, 22, 20, 30, 34, 6, 14, 6, 58, 6, 30, 6, 8, 52, 8, 30, 40, 6, 66, 20, 40, 50, 10, 48, 12, 8, 36, 84, 6, 6, 24, 84, 40, 6, 66, 14, 24

Offset: 1

Labos Elemer, Mar 06 2000

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A031924.

Cf. A053322, A053323, A053324, A053325, A053326, A053327, A053331.

P:=Filtered([1..2100],IsPrime);;
P1:=List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]);;
a:=List([1..Length(P1)-1],i->P1[i+1]-P1[i]);; Print(a); # Muniru A Asiru, Dec 23 2018

With[{p = Prime[Range[330]]}, Differences[p[[Position[Differences[p], 6] // Flatten]]]] (* Amiram Eldar, Mar 10 2025 *)

A053322 First differences of A031926.

Original entry on oeis.org

270, 30, 12, 48, 30, 12, 192, 18, 18, 24, 18, 150, 18, 54, 126, 54, 30, 180, 66, 84, 36, 12, 162, 90, 156, 24, 150, 60, 30, 30, 186, 72, 78, 54, 36, 42, 102, 36, 30, 102, 30, 168, 12, 228, 42, 132, 78, 18, 162, 408, 60, 234, 168, 192, 108, 120, 18, 210, 174, 120, 90

Offset: 1

Labos Elemer, Mar 06 2000

nonn

Minimal value 12 is for n = 3, 6, 22, 43, 90, 123, 125, 135, 144, 147, 201, 255, 276, 287, 310, 338, 350. - Zak Seidov, Jun 12 2017

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

Cf. A031926.

Cf. A053321, A053323, A053324, A053325, A053326, A053327, A053331.

With[{p = Prime[Range[1000]]}, Differences[p[[Position[Differences[p], 8] // Flatten]]]] (* Amiram Eldar, Mar 10 2025 *)

A053324 First differences of A031930.

Original entry on oeis.org

12, 256, 42, 110, 42, 136, 200, 204, 36, 70, 152, 40, 12, 20, 178, 80, 22, 78, 180, 30, 198, 102, 48, 132, 42, 156, 150, 122, 18, 102, 22, 68, 72, 16, 152, 60, 100, 272, 58, 90, 20, 298, 12, 140, 130, 12, 110, 76, 42, 120, 48, 110, 64, 158, 88, 320, 100, 174, 50

Offset: 1

Labos Elemer, Mar 06 2000

nonn

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

Cf. A031930.

Cf. A053321, A053322, A053323, A053325, A053326, A053327, A053331.

Differences[Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==12&][[;;,1]]] (* Harvey P. Dale, Sep 28 2024 *)

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 *)

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 *)

A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

Original entry on oeis.org

409, 1039, 2017, 2719, 3571, 4219, 4231, 4261, 4327, 6079, 6121, 6679, 6781, 8209, 11047, 11149, 11959, 12241, 15277, 19531, 19687, 21577, 21589, 26881, 27529, 28087, 28297, 29389, 30829, 30859, 31069, 32401, 42061, 45307, 47797, 48109

Offset: 1

Labos Elemer, Mar 22 2000

nonn

Subsequence of lesser terms of 10-twins (A031928).
Start primes of quadruples consisting of two consecutive 10-twins of prime which are in minimal distance [minD = A052380(10/2) = 12].
First term of this sequence is 409 = A052381(5).

			p=1039 begins [1039,1049,1051,1061] prime quadruple with the appropriate difference pattern: [10,2,10]=[d,D-d,d], so d=10, D=12.

Robert G. Wilson v, Table of n, a(n) for n = 1..1780

Cf. A031928, A053323, A052380, A052381, A052377, A052378.

{p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, p]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)

a(n)=p, a prime which begins a [p, p+d, p+D, p+D+d]=[p, p+10, p+12, p+22] prime quadruple.

a(n) = A259025(n)-11. - Robert G. Wilson v, Jul 15 2015

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A031928 Lower prime of a difference of 10 between consecutive primes.

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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.

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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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A053325 First differences of A031932.

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A053321 First differences of A031924.

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A053322 First differences of A031926.

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A053324 First differences of A031930.

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A053326 First differences of A031934.

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