A078858 -id:A078858 - cp's OEIS frontend

A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

67, 1447, 2377, 2707, 5437, 5737, 7207, 9337, 11827, 12037, 19207, 21487, 21517, 23197, 26107, 26947, 28657, 31147, 31177, 35797, 37357, 37567, 42697, 50587, 52177, 65167, 67927, 69997, 71707, 74197, 79147, 81547, 103087, 103387, 106657

Offset: 1

Labos Elemer, Dec 11 2002

nonn

Subsequence of A022005. - R. J. Mathar, May 06 2017

			p=67,67+4=71,67+4+2=73,67+4+2+6=79 are consecutive primes.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous prime quadruple sequences with various possible {2, 4, 6}-difference-patterns in brackets: A007530[242], A078847[246], A078848[264], A078849[266], A052378[424], A078850[426], A078851[462], A078852[466], A078853[624], A078854[626], A078855[642], A078856[646], A078857[662], A078858[664], A033451[666].

d = {4, 2, 6}; First /@ Select[Partition[Prime@ Range@ 12000, Length@ d + 1, 1], Differences@ # == d &] (* Michael De Vlieger, May 02 2016 *)

Primes p = p(i) such that p(i+1)=p+4, p(i+2)=p+4+2, p(i+3)=p+4+2+6.

Listed terms verified by Ray Chandler, Apr 20 2009

A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

Original entry on oeis.org

7, 13, 31, 37, 73, 97, 157, 223, 373, 433, 1087, 1291, 1423, 1483, 1543, 1861, 1987, 2341, 2383, 2677, 2683, 3313, 3607, 4441, 4507, 4783, 4993, 5641, 5851, 6037, 6961, 7237, 7867, 8731, 9613, 9733, 10723, 13093, 13681, 14143, 14731, 16057, 16411, 16921, 17377

Offset: 1

Alexander Yutkin, Apr 05 2025

nonn

The four primes need not be consecutive; otherwise we have the sequence A078856.

			p=37: 37+6=43, 37+10=47, 37+16=53 -> prime quartet: (37, 43, 47, 53).

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000040, A001223, A023200, A031924.

Cf. A078852 [4, 6, 6], A078856 [6, 4, 6], A078858 [6, 6, 4], A033451 [6, 6, 6].

q:= p-> andmap(i->isprime(p+i), [0, 6, 10, 16]):
select(q, [$2..20000])[];  # Alois P. Heinz, Apr 05 2025

Select[Prime[Range[2000]],AllTrue[#+{6,10,16},PrimeQ]&] (* James C. McMahon, Apr 13 2025 *)

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Original entry on oeis.org

3, 7, 15, 26, 38, 48, 67, 92, 105, 108, 109, 118, 130, 128, 112, 80, 36, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Dec 19 2002

nonn

The ">n+1" rules out n-tuples like (2,2), which only occurs for the primes 3, 5, 7. All terms from a(19) on equal 0.

An n-tuple (a_1,a_2,...,a_n) is counted iff the partial sums 0, a_1, a_1+a_2, ..., a_1+...+a_n do not contain a complete residue system (mod p) for any prime p.

Eric Weisstein's World of Mathematics, k-Tuple Conjecture

The 26 4-tuples and 38 5-tuples are in A078868 and A078870. Cf. A001359, A008407, A029710, A031924, A022004-A022007, A078852, A078858, A078946-A078969, A020497.

test[tuple_] := Module[{r, sums, i, j}, r=Length[tuple]; sums=Prepend[tuple.Table[If[j>=i, 1, 0], {i, 1, r}, {j, 1, r}], 0]; For[i=1, Prime[i]<=r+1, i++, If[Length[Union[Mod[sums, Prime[i]]]]==Prime[i], Return[False]]]; True]; tuples[0]={{}}; tuples[n_] := tuples[n]=Select[Flatten[Outer[Append, tuples[n-1], {2, 4, 6}, 1], 1], test]; a[n_] := Length[tuples[n]]

Edited by Dean Hickerson, Dec 20 2002

A078966 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,4,2).

Original entry on oeis.org

601, 2671, 20341, 24091, 41941, 42391, 55201, 65701, 87541, 125101, 198811, 249421, 355501, 414691, 416401, 428551, 510061, 521161, 541531, 543871, 560221, 603901, 609601, 637711, 663961, 669661, 743161, 770041, 986131, 1020961, 1026661, 1099711, 1113181, 1120501

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+16 and p+18 are consecutive primes.

			601 is in the sequence since 601, 607 = 601 + 6, 613 = 601 + 12, 617 = 601 + 16 and 619 = 601 + 18 are consecutive primes.

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

Subsequence of A078858. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Transpose[Select[Partition[Prime[Range[81000]],5,1],Differences[#] == {6,6,4,2}&]][[1]] (* Harvey P. Dale, Sep 15 2011 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

a(n) == 1 (mod 30). - Amiram Eldar, Feb 22 2025

Edited by Dean Hickerson, Dec 20 2002

A078967 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,4,6).

Original entry on oeis.org

151, 367, 3307, 4987, 20101, 30097, 52951, 53617, 85831, 92221, 95701, 99817, 103561, 128461, 135601, 163621, 214651, 221071, 241321, 241861, 246907, 274831, 280591, 282691, 287851, 294787, 295831, 297601, 307261, 308311, 334771, 340897, 347161, 350431, 354301

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+16 and p+22 are consecutive primes.

			151 is in the sequence since 151, 157 = 151 + 6, 163 = 151 + 12, 167 = 151 + 16 and 173 = 151 + 22 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A078858. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078971, A022006, A022007.

Transpose[Select[Partition[Prime[Range[30000]],5,1],Differences[#] == {6,6,4,6}&]][[1]] (* Harvey P. Dale, Apr 06 2012 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 22 2025

From Amiram Eldar, Feb 22 2025: (Start)
a(n) == 1 (mod 6).
a(n) == 1 or 7 (mod 30). (End)

Edited by Dean Hickerson, Dec 20 2002

A079018 Suppose p and q = p+16 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 17 possible difference patterns, namely [16], [4,12], [6,10], [10,6], [12,4], [4,2,10], [4,6,6], [4,8,4], [6,4,6], [6,6,4], [10,2,4], [4,2,4,6], [4,2,6,4], [4,6,2,4], [6,4,2,4], [4,2,4,2,4], [2,2,4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 7, 13, 31, 43, 67, 73, 151, 181, 211, 241, 277, 331, 463, 487, 1597, 1831

Offset: 1

Labos Elemer, Jan 24 2003

			p=181, q=197 has difference pattern [10,2,4] and {181,191,193,197} is the corresponding consecutive prime 4-tuple.

A022008(1)=7, A078952(1)=13, A078852(1)=73, A078953(1)=67, A078954(1)=1597, A078961(1)=31, A078856(1)=73, A078858(1)=151, A031934(1)=A000230(8)=1831.

Cf. A079016-A079024.

A215719 The smallest of four consecutive primes with prime gaps {a,b,c} = {10,18,2}.

Original entry on oeis.org

1249, 14293, 17929, 31741, 32089, 33151, 35869, 57193, 60859, 64891, 71443, 85303, 87481, 90793, 93103, 98533, 99679, 99961, 108079, 131221, 135319, 139429, 140731, 144451, 157639, 165559, 171439, 175909, 180043, 186619, 193153, 203353, 214531, 217489

Offset: 1

V.J. Pohjola, Aug 22 2012

nonn

Conjecture: The terms of any feasible prime gap triple {a,b,c} to form a quadruple of consecutive primes are sums of terms of three consecutive subsequences of the infinite integer sequence with period (4,2,4,2,4,6,2,6). By this token all possible sequences of quadruples of consecutive primes can be generated, including those already in the OEIS.

			The terms of the prime gap triple {10,18,2} are the sums of the terms of the following (arbitrarily chosen) subsequences ..., {4,2,4}, {6,2,6,4}, {2}, ... For n=3, a(n) = 17929 is the smallest prime of the third prime quadruple {17929, 17939, 17957, 17959}.

Robert Israel, Table of n, a(n) for n = 1..4147

Cf. A078858.

N:= 10^6; # to get all terms <= 6*N
Primes1:= select(isprime,{seq(6*i+1,i=1..N+5)}):
Primes5:= select(isprime,{seq(6*i+5,i=1..N+5)}):
Q:= `intersect`(Primes1, map(t->t-10, Primes5), map(t->t-28,Primes5), map(t->t-30,Primes1):
A215719:= select(t -> select(isprime,{seq(t+2*i,i=1..13)}) = {t+10}, Q): # Robert Israel, May 04 2014

Definition and comment corrected by Robert Israel, May 04 2014

cp's OEIS Frontend

A078850 Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A382810 Primes p such that p + 6, p + 10 and p + 16 are also primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A078869 Number of n-tuples with elements in {2,4,6} which can occur as the differences between n+1 consecutive primes > n+1. (Values of a(11), ..., a(18) are conjectured to be correct, but are only known to be upper bounds.)

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A078966 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,4,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A078967 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,4,6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

Views

Author

Keywords

Examples

Crossrefs

A215719 The smallest of four consecutive primes with prime gaps {a,b,c} = {10,18,2}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions