A031928 -id:A031928 - cp's OEIS frontend

A053323 First differences of A031928.

42, 60, 42, 54, 72, 12, 126, 30, 54, 60, 18, 78, 24, 18, 90, 102, 18, 12, 102, 18, 78, 150, 72, 156, 72, 24, 78, 78, 138, 12, 24, 36, 54, 378, 126, 72, 12, 36, 120, 30, 84, 108, 252, 156, 30, 24, 12, 126, 60, 54, 30, 348, 18, 12, 12, 18, 12, 54, 12, 24, 120, 180, 198, 48

Offset: 1

Labos Elemer, Mar 06 2000

Minimal value is 12; a(n) = 12 for n = 6, 22, 128, 172, 218, 229, 248, 253, 320, 344. - Zak Seidov, Jun 12 2017

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

Cf. A031928, A079033.

Cf. A053321, A053322, A053324, A053325, A053326, A053327, A053331.

Differences[Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==10&][[All,1]]] (* Harvey P. Dale, Jan 16 2017 *)

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

A086135 Numbers n such that n and n+10 are both prime but are non-consecutive; which means that at least one prime is between n and n+10; it is not identical with A023303 because here the terms of A031928 are missing.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 73, 79, 97, 103, 127, 157, 163, 223, 229, 271, 307, 349, 373, 379, 433, 439, 457, 499, 607, 643, 673, 733, 751, 853, 877, 937, 967, 1009, 1087, 1093, 1213, 1279, 1291, 1297, 1423, 1429, 1483, 1489, 1543, 1549, 1597, 1609, 1657, 1777

Offset: 1

Labos Elemer, Jul 28 2003

nonn

			First deviation from A023303 = {3,7,13,19,31,37,43,61,73,79,97,103,127,139,157,..} is due to the absence of 139=A031928(1).

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

Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&& !Equal[s1-s, d], Print[s]], {n, 1, 1000}]
Select[Prime[Range[300]],PrimeQ[#+10]&&NextPrime[#]!=(#+10)&] (* Harvey P. Dale, Oct 25 2020 *)

Complement of a=A031928 with respect to b=A023303: [b]&[nota]: this and A031928 are disjoint, but A031928 is a proper subset of A023303.

A023203 Primes p such that p + 10 is also prime.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967, 1009, 1021, 1039, 1051

Offset: 1

David W. Wilson

A subset of A002476. It appears that this is also a subset of A007645. The first few terms of A007645 that are not in this sequence are {67, 109, 151, 193, 199, 211, 277, 313, 331, 367, 397, 463, 487, 523, 541, 571, 601, 613, ...}. - Alexander Adamchuk, Aug 15 2006

The entries are all in A007645, because they cannot be of the form p = 3*j + 2. If they were, p + 10 = 3*j + 12 would be divisible by 3 and not prime. - R. J. Mathar, Oct 30 2009

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Twin Primes

Different from A015916. Cf. A031928, A079033.

Cf. A002476, A007645, A092146.

[n: n in [0..1000] | IsPrime(n) and IsPrime(n+10)]; // Vincenzo Librandi, Nov 20 2010

for p from 1 to 10000 do if isprime(p) and isprime(p+10) then print(p) end if end do # Matt C. Anderson, Aug 26 2022

Select[Prime[Range[200]], PrimeQ[# + 10] &] (* Harvey P. Dale, Dec 14 2011 *)

is(n)=isprime(n)&&isprime(n+10) \\ Charles R Greathouse IV, Jul 01 2013

Revised by N. J. A. Sloane, Jan 29 2013

New name from Michel Marcus, Mar 04 2020

A031930 Lower prime of a difference of 12 between consecutive primes.

Original entry on oeis.org

199, 211, 467, 509, 619, 661, 797, 997, 1201, 1237, 1307, 1459, 1499, 1511, 1531, 1709, 1789, 1811, 1889, 2069, 2099, 2297, 2399, 2447, 2579, 2621, 2777, 2927, 3049, 3067, 3169, 3191, 3259, 3331, 3347, 3499, 3559, 3659, 3931, 3989

Offset: 1

Jeff Burch

nonn

Some of the terms of this sequence are primes that are separated from both their predecessor and successor primes by 12, e.g., 211, 1511, 4409, 4691, 7841, 9871, 11299, 11411, 11731.  See A053072. - Harvey P. Dale, Apr 07 2013
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 comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014
Aside from 2 and 3, all primes are congruent to 1, 5, 7, 11 mod 12. Thus the least significant duodecimal digit of any term in this sequence is 1, 5, 7 or B. - Alonso del Arte, Aug 19 2017

			199 is a term as the next prime is 199 + 12 = 211.
211 is also a term since the next prime is 211 + 12 = 223.
But 223 is not a term since the next prime is 227, and 223 + 12 = 235 = 5 * 47.

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

Cf. A053072, A248855, A031928.

[p: p in PrimesUpTo(4000) | NextPrime(p)-p eq 12]; // Bruno Berselli, Apr 09 2013

Transpose[Select[Partition[Prime[Range[600]], 2, 1], Last[#] - First[#] == 12 &]][[1]] (* Harvey P. Dale, Apr 07 2013 *)

is(n)=nextprime(n+1)==n+12 && isprime(n) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = prime(A320704(n)). - R. J. Mathar, Apr 30 2024

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

A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

Original entry on oeis.org

3, 7, 47, 389, 409, 199, 24749, 3373, 20183, 46703, 19867, 16763, 142811, 14563, 69593, 763271, 276637, 255767, 363989, 383179, 247099, 2130809, 15370423, 3565931, 458069, 9401647, 6314393, 20823437, 9182389, 4911251, 15442121

Offset: 1

Labos Elemer, Mar 13 2000

nonn

A prime quadruple (triple), {[p,p+d],[p+D,p+D+d]} is called a "non-overlapping" (disjoint or touching) pair of twins if D = distance >= d = difference "inside" twin.

			If n=23, d=46, min{D}=48 then the first suitable quadruple of primes is [15370423, 15370469, 15370471, 15370517] with difference pattern [46, 2, 46]; if n=3, d=6, min{D}=6 then the first such triple is [47, 53, 53, 59] = [47, 53, 59] with difference pattern [6, 6].

The first 10 terms here appear as initial terms in A052350-A052359.

Smallest p so that [p, p+2n], [p+min{D}, p+2n+min{D}] is a quadruple (or triple if d=min{D}) of consecutive primes.

Corrected by Jud McCranie, Jan 04 2001

a(11) corrected by Sean A. Irvine, Nov 07 2021

A098974 Primes p such that q-p = 24, where q is the next prime after p.

Original entry on oeis.org

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

A126784 Primes p such that q-p = 32, where q is the next prime after p.

Original entry on oeis.org

5591, 10799, 27701, 27851, 33647, 39047, 41081, 41687, 43721, 44417, 45989, 47459, 50789, 52457, 55259, 55547, 61781, 62351, 64817, 66239, 67307, 69959, 73907, 79907, 80567, 82307, 84089, 88037, 94169, 94961, 99191, 99929, 100559, 102611

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Feb 24 2007

Lower prime of a difference of 32 between consecutive primes.

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

Cf. A001359, A029710, A031924, A031926, A031928, A031930, A031932, A031934, A031936, A031938, A061779, A098974, A124594, A124595, A124596, A000230, A050434.

lista(nn) = {p = 2; while (p < nn, q = nextprime(p+1); if (q - p == 32, print1(p, ", ")); p = q;);} \\ Michel Marcus, Jul 17 2013

A083371 Primes p such that q-p >= 8, where q is the next prime after p.

Original entry on oeis.org

89, 113, 139, 181, 199, 211, 241, 283, 293, 317, 337, 359, 389, 401, 409, 421, 449, 467, 479, 491, 509, 523, 547, 577, 619, 631, 661, 683, 691, 701, 709, 719, 743, 761, 773, 787, 797, 811, 829, 839, 863, 887, 911, 919, 929, 953, 983, 997, 1021, 1039, 1051, 1069

Offset: 1

Benoit Cloitre, Jun 04 2003

nonn

The original definition by Cloitre was: [Start from any initial value F(1) >= 2 and define F(n) as the largest prime factor of F(1)+F(2)+F(3)+...+F(n-1). The sequence contains the primes satisfying F(2*p)=p supposed F(1)=7. Conjecture: F(n)= n/2+O(log n) and the sequence is infinite.] Don Reble showed Jan 22 2022 that these are the same primes p followed by a prime gap of q-p >=8, where q is the next prime after p: [
Let X' be the first prime after X, 'X be the first prime before X.
The F sequence starting at "7" has 11 "7"s, then 6 "11"s, 6 "13"s, 6 "17"s, 6 "19"s, 10 "23"s, ...
One easily sees that the F sequence starting at prime S has S' instances of S; then for each prime P after S, it has (P'-'P) instances of P.  (A076973 is the F sequence starting at "2".)
The primes from S to P occupy the first [S' + (S''-S) + (S'''-S') + ... + (P' - 'P)] terms of F.
That sum telescopes to P'+P-S, and so
    F(P'+P-S) = P;  F(P'+P-S+1) = P';
    F(P+'P-S) = 'P; F(P+'P-S+1) = P.
If F(X) =P, then P+'P-S < X   <= P'+P-S.
If F(2P)=P, then P+'P-S < 2P  <= P'+P-S
                     'P < P+S <= P'
                            S <= P'-P
So this sequence has the primes P for which P'-P >= 7; and since P'-P is even (both primes are odd), P'-P >= 8. q.e.d.]

Nathaniel Johnston, Table of n, a(n) for n = 1..1500
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A076973.

d:=8; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0; # N. J. A. Sloane, Dec 19 2006
f := proc(n) option remember: if(n=1)then return 7: fi: return max(op(numtheory[factorset](add(f(i),i=1..n-1)))): end: seq(`if`(f(2*ithprime(n))=ithprime(n),ithprime(n),NULL),n=1..200); # Nathaniel Johnston, Jun 25 2011, via Cloitre's F

Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]>7&]][[1]] (* Harvey P. Dale, Jan 28 2013 *)

A000040 MINUS A124590. - R. J. Mathar, Jan 23 2022

A031926 UNION A031928 UNION A031930 UNION A031932 UNION ... - R. J. Mathar, Jan 23 2022

Terms after a(20) from Nathaniel Johnston, Jun 26 2011

Merged with A124583 in response to Reble's seqfan post. - R. J. Mathar, Jan 24 2022

cp's OEIS Frontend

A053323 First differences of A031928.

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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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A086135 Numbers n such that n and n+10 are both prime but are non-consecutive; which means that at least one prime is between n and n+10; it is not identical with A023303 because here the terms of A031928 are missing.

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A023203 Primes p such that p + 10 is also prime.

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Magma

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A031930 Lower prime of a difference of 12 between consecutive primes.

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Magma

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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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A052381 The smallest initial prime of 2 non-overlapping d-twin primes if the distance between pairs (D) is minimal (see A052380).

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A098974 Primes p such that q-p = 24, where q is the next prime after p.

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