A052380 -id:A052380 - cp's OEIS frontend

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

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

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

A264788 a(n) is the number of circles added at n-th iteration of the pattern starting with 2 circles. (See comment.)

Original entry on oeis.org

2, 2, 4, 4, 6, 10, 10, 12, 16, 16, 18, 22, 22, 24, 28, 28, 30, 34, 34, 36, 40, 40, 42, 46, 46, 48, 52, 52, 54, 58, 58, 60, 64, 64, 66, 70, 70, 72, 76, 76, 78, 82, 82, 84, 88, 88, 90, 94, 94, 96, 100, 100, 102, 106, 106, 108, 112, 112, 114, 118, 118, 120, 124

Offset: 0

Kival Ngaokrajang, Nov 25 2015

Pattern construction rules: (i) At n = 0, there are two circles of radius s with centers at the ends of a straight line of length s. (ii) At n > 0, draw circles by placing center at the intersection points of the circumferences of circles in the previous iteration, with overlaps forbidden. The pattern seems to be the flower of life. See illustration.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Sacred Geometry, Flower of Life
Eric Weisstein's World of Mathematics, Flower of Life
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A052380, A136289.

LinearRecurrence[{1,0,1,-1},{2,2,4,4,6,10},100] (* Paolo Xausa, Nov 17 2023 *)

{a = 4; print1("2, 2, ", a, ", "); for(n = 2, 100, if (Mod(n,3)==0, d1 = 2); if (Mod(n,3)==1, d1 = 4);  if (Mod(n,3)==2, d1 = 0); a = a + d1; print1(a, ", "))}

Vec(2*(1+x^2-x^3+x^4+x^5)/((1-x)^2*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Dec 10 2015

From Colin Barker, Dec 10 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
G.f.: 2*(1+x^2-x^3+x^4+x^5) / ((1-x)^2*(1+x+x^2)).
(End)

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511

Offset: 1

Labos Elemer, Mar 22 2000

nonn

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.

A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

Original entry on oeis.org

11, 29, 59, 641, 101, 347, 2309, 569, 1931, 521, 1787, 419, 1229, 1871, 3671, 2237, 6551, 1427, 21491, 1607, 12377, 4931, 1019, 23201, 809, 19697, 12539, 2549, 38921, 10709, 37547, 8819, 9239, 34031, 6089, 80447, 15581, 46049, 36341, 14867, 38237, 36779, 87509, 71261, 15137, 40427, 13679, 54917, 41141, 50891

Offset: 1

Artur Jasinski, Jan 12 2021

nonn

Lesser twin primes (with the exception of prime 3) are congruent to 5 modulo 6, which implies that distances between successive pairs of twin primes are 6*k.

			a(1)=11 because 11 - 5 = 6*1.
a(2)=41 because 41 - 29 = 6*2.
a(3)=59 because 59 - 41 = 6*3.

Martin Raab, Table of n, a(n) for n = 1..4508

Cf. A001359, A001359, A053319, A007530, A052380, A052381, A113274, A113275, A052350.

Table[a[n] = 0, {n, 1, 10000}]; Table[
b[n] = 0, {n, 1, 10000}]; qq = {}; prev = 5; Do[
If[Prime[n + 1] - Prime[n] == 2, k = (Prime[n] - prev)/6;
  If[b[k] == 0, a[k] = Prime[n]; b[k] = 1]; prev = Prime[n]], {n, 5,
  10000}]; list = Table[a[n], {n, 1, 50}]
(* Second program: *)
pp = Select[Prime[Range[10^4]], PrimeQ[#+2]&];
dd = Differences[pp];
a[n_] := pp[[FirstPosition[dd, 6n][[1]]+1]];
Array[a, 50] (* Jean-François Alcover, Jan 13 2021 *)

a(n) = A052350(n) + 6*n.

cp's OEIS Frontend

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

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

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

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A264788 a(n) is the number of circles added at n-th iteration of the pattern starting with 2 circles. (See comment.)

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A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

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A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

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