A052376 -id:A052376 - cp's OEIS frontend

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

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].

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

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.

A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.

Original entry on oeis.org

420, 1050, 2028, 2730, 3582, 4230, 4242, 4272, 4338, 6090, 6132, 6690, 6792, 8220, 11058, 11160, 11970, 12252, 15288, 19542, 19698, 21588, 21600, 26892, 27540, 28098, 28308, 29400, 30840, 30870, 31080, 32412, 42072, 45318, 47808, 48120

Offset: 1

Karl V. Keller, Jr., Jun 16 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are congruent to 0 mod 30.
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.

			For n=420: 409, 419, 421, 431 are consecutive primes (n-11=409, n-1=419, n+1=421, n+11=431).
For n=1050: 1039, 1049, 1051, 1061 are consecutive primes (n-11=1039, n-1=1049, n+1=1051, n+11=1061).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A052376, A077800 (twin primes), A014574, A249674 (30n), A256753.

{p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, q + 1]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)
Mean/@Select[Partition[Prime[Range[5000]],4,1],Differences[#]=={10,2,10}&] (* Harvey P. Dale, Sep 11 2019 *)

is(n)=n%6==0&&isprime(n-11)&&isprime(n-1)&&isprime(n+1)&&isprime(n+11)&&!isprime(n-7)&&!isprime(n-5)&&!isprime(n+5)&&!isprime(n+7) \\ Charles R Greathouse IV, Jul 17 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,50001,2):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-11 and nextprime(i+1) == i+11 :  print (i,end=', ')

a(n) = A052376(n) + 11. - Robert G. Wilson v, Jul 15 2015

cp's OEIS Frontend

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

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

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A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.

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