A053319 -id:A053319 - cp's OEIS frontend

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

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

A309066 Integers m such that A053319(m) is a term in A014574.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 51, 53, 57, 59, 61, 63, 65, 66, 68, 69, 70, 71, 73, 75, 77, 78, 79, 81, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Dmitry Kamenetsky, Jul 10 2019

nonn

			A053319(2) = 6 and 6 is in A014574, so a(1) = 2.
A053319(3) = 6 and 6 is in A014574, so a(2) = 3.
A053319(4) = 12 and 12 is in A014574, so a(3) = 4.

Cf. A014574, A053319, A309067.

tpmidQ[n_] := And @@ PrimeQ[n + {-1, 1}]; ind = 0; tp = {4}; s = {}; Do[If[ tpmidQ[n], ind++; If[tpmidQ[n - tp[[-1]]], AppendTo[s, ind]]; AppendTo[tp, n]], {n, 6, 10^4}]; s (* Amiram Eldar, Jul 12 2019 *)

A309067 Index of A053319(n) in A014574; -1 if it doesn't exist.

Original entry on oeis.org

-1, 2, 2, 3, 3, 4, 3, 5, 2, 5, 3, 5, 3, 2, 5, 3, 5, 3, 5, -1, 8, 3, 5, 7, -1, 5, 4, -1, 4, 12, 3, 2, 5, -1, 11, 3, 4, 3, 5, 7, -1, -1, 3, 3, 4, 10, -1, 5, 2, -1, 3, -1, 5, -1, -1, -1, 2, -1, 4, -1, 5, -1, 2, -1, 4, 3, -1, 5, 6, 5, 6, -1, 6, -1, 5, -1, 4, 7, 3, -1, 5, -1, -1, -1, 2, 6, 5, 5, 3, 4

Offset: 1

Dmitry Kamenetsky, Jul 10 2019

sign

Cf. A014574, A053319, A309066.

ind[v_, e_] := If[Length[(p = Position[v, ?(# == e &)])] == 0, -1, p[[1]]]; tpmidQ[n] := And @@ PrimeQ[n + {-1, 1}]; tp = Select[Range[10^4], tpmidQ]; d = Differences[tp]; ind[tp, #] & /@ d // Flatten (* Amiram Eldar, Jul 12 2019 *)

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

A144834 Numbers n such that the two numbers n+1 and n+3 are both prime.

Original entry on oeis.org

2, 4, 10, 16, 28, 40, 58, 70, 100, 106, 136, 148, 178, 190, 196, 226, 238, 268, 280, 310, 346, 418, 430, 460, 520, 568, 598, 616, 640, 658, 808, 820, 826, 856, 880, 1018, 1030, 1048, 1060, 1090, 1150, 1228, 1276, 1288, 1300, 1318, 1426, 1450, 1480, 1486, 1606

Offset: 1

Giovanni Teofilatto, Sep 22 2008

1 less than the lesser of each twin prime pair. [Harvey P. Dale, Nov 08 2011]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A053319 (first differences).

Transpose[Select[Partition[Prime[Range[300]],2,1],Last[#]-First[#] == 2&]][[1]]-1 (* Harvey P. Dale, Nov 08 2011 *)

a(n) = A014574(n)-2 = A001359(n)-1. - R. J. Mathar, Sep 24 2008

Definition edited and extended by R. J. Mathar, Sep 24 2008

A270535 Integers k such that A001359(k) + A001359(k+3) = A001359(k+1) + A001359(k+2).

Original entry on oeis.org

5, 8, 10, 11, 15, 16, 17, 27, 36, 68, 69, 71, 111, 132, 189, 200, 212, 214, 234, 252, 262, 279, 317, 332, 343, 344, 364, 424, 426, 500, 506, 518, 520, 543, 563, 577, 606, 620, 658, 672, 696, 697, 737, 766, 882, 907, 982, 1009, 1064, 1087, 1089, 1091, 1162, 1164, 1172, 1226, 1256, 1268

Offset: 1

Altug Alkan, Mar 18 2016

nonn

Integers k such that A006512(k) + A006512(k+3) = A006512(k+1) + A006512(k+2).

Integers k such that A014574(k) + A014574(k+3) = A014574(k+1) + A014574(k+2).

			5 is a term because A001359(5) = 29, A001359(6) = 41, A001359(7) = 59, A001359(8) = 71 and 29 + 71 = 41 + 59.

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

Cf. A001359, A006512, A014574, A053319.

s = Select[Prime@ Range[10^6], PrimeQ[# + 2] &]; Select[Range@ 1300, s[[#]] + s[[# + 3]] == s[[# + 1]] + s[[# + 2]] &] (* after Robert G. Wilson v at A001359 *)

t(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
b(n) = t(n) + t(n+3) - t(n+1) - t(n+2);
for(n=1, 2000, if(b(n) == 0, print1(n, ", ")));

list(lim) = {my(k = 0, p1 = 2, t = [0, 0, 0, 0]); forprime(p2 = 3, lim, if(p2 - p1 == 2, k++; t = concat(t[2..4], p1); if(t[1] + t[4] == t[2] + t[3], print1(k-3, ", "))); p1 = p2);} \\ Amiram Eldar, Feb 22 2025

A175668 First differences of A175648.

Original entry on oeis.org

4, 11, 1, 12, 1, 16, 7, 7, 17, 5, 4, 20, 4, 3, 1, 10, 12, 1, 13, 28, 18, 1, 3, 4, 4, 1, 1, 2, 32, 25, 13, 4, 4, 3, 1, 2, 4, 14, 4, 12, 23, 3, 16, 5, 9, 3, 9, 4, 4, 2, 34, 7, 15, 9, 3, 4, 4, 4, 4, 4, 10, 4, 14, 4, 5, 24, 17, 43, 7, 38, 14, 4, 9, 1, 4, 4, 10, 4, 28, 4, 14, 4, 14, 4, 4, 10, 4, 10

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2010

nonn

Distance between twin semiprime pairs.

Cf. A053319, A175612, A175648.

A175648 := proc(n) option remember; if n = 1 then 6; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and numtheory[bigomega](a+4) = 2 then return a; end if; end do: end if; end proc:
A175668 := proc(n) A175648(n+1)-A175648(n) ; end proc:
seq(A175668(n),n=1..100) ; # R. J. Mathar, Aug 07 2010

Terms from a(33) on corrected by R. J. Mathar, Aug 07 2010

A175683 Numbers n such that 30n-13, 30n-11, 30n-1, 30n+1, 30n+11, 30n+13 are all prime.

Original entry on oeis.org

1, 43, 141, 720, 3038, 3466, 3772, 4068, 7896, 11402, 14070, 17499, 18683, 20887, 25166, 26586, 30311, 33237, 44072, 49791, 56629, 58268, 58764, 71483, 71953, 74284, 79939, 86022, 87199, 88941, 91951, 92273, 100176, 102019, 107505, 109438

Offset: 1

Juri-Stepan Gerasimov, Aug 09 2010

nonn

			a(1)=1 because 30*1-13=17=prime, 30*1-11=19=prime, 30*1-1=29=prime, 30*1+1=31=prime, 30*1+11=41=prime, 30*1+13=43=prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A053319.

Corrected (3472 replaced by 3772, 49776 replaced by 49791) and extended by R. J. Mathar, Aug 13 2010

A270541 a(n) = A001359(n) - A001359(n+1) - A001359(n+2) + A001359(n+3).

Original entry on oeis.org

4, 6, 6, 6, 0, 12, -6, 0, 6, 0, 0, -24, 18, 6, 0, 0, 0, 24, 42, -24, -42, 48, 18, -30, -30, -6, 0, 126, -6, -144, 18, 18, 108, -12, -120, 0, 12, 48, 48, -12, -66, -36, 6, 96, 6, -78, -18, 90, 6, -72, 18, -24, 36, 60, -60, -30, 12, -6, 12, 6, -24, -30, 12, -12, 78, 18, -54, 0, 0, 138, 0, -102, -12, -42

Offset: 1

Altug Alkan, Mar 18 2016

sign

6*k appears for the form of a(n) for n > 1.
What is the most repeated value of a(n)?
See A270535 for the position of 0's in this sequence.

			a(1) = 4 because a(1) = A001359(1) - A001359(2) - A001359(3) + A001359(4) = 3 - 5 - 11 + 17 = 4.

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

Cf. A001359, A006512, A014574, A053319.

s = Select[Prime@Range[10^6], PrimeQ[# + 2] &]; Table[s[[n]] - s[[n + 1]] - s[[n + 2]] + s[[n + 3]], {n, 74}] (* Michael De Vlieger, Mar 19 2016, after Robert G. Wilson v at A001359 *)
#[[1]]-#[[2]]-#[[3]]+#[[4]]&/@Partition[Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]]==2&][[;;,1]],4,1] (* Harvey P. Dale, Jun 14 2025 *)

t(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
a(n) = t(n) + t(n+3) - t(n+1) - t(n+2);
for(n=1, 200, print1(a(n), ", "));

a(n) = A053319(n+2) - A053319(n).

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

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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A309066 Integers m such that A053319(m) is a term in A014574.

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A309067 Index of A053319(n) in A014574; -1 if it doesn't exist.

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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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A144834 Numbers n such that the two numbers n+1 and n+3 are both prime.

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A270535 Integers k such that A001359(k) + A001359(k+3) = A001359(k+1) + A001359(k+2).

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A175668 First differences of A175648.

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A175683 Numbers n such that 30n-13, 30n-11, 30n-1, 30n+1, 30n+11, 30n+13 are all prime.

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