A078946 -id:A078946 - cp's OEIS frontend

A078866 The quadruples (d1,d2,d3,d4) with elements in {2,4,6} are listed in lexicographic order; for each quadruple, this sequence lists the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), if such a prime exists.

5, 17, 41, 29, 71, 149, 3299, 7, 13, 67, 1597, 19, 43, 12637, 1601, 23, 593, 31, 61, 3313, 157, 47, 601, 151, 251, 3301

Offset: 1

Labos Elemer, Dec 19 2002

The 26 quadruples for which p exists are listed, in decimal form, in A078868.

			The term 12637 corresponds to the quadruple (4,6,6,6): 12637, 12641, 12647, 12653 and 12659 are consecutive primes.

The quadruples are in A078868. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Edited by Dean Hickerson, Dec 20 2002

A078867 Sorted version of A078866.

Original entry on oeis.org

5, 7, 13, 17, 19, 23, 29, 31, 41, 43, 47, 61, 67, 71, 149, 151, 157, 251, 593, 601, 1597, 1601, 3299, 3301, 3313, 12637

Offset: 1

Labos Elemer, Dec 19 2002

Each term is the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), for some quadruple (d1,d2,d3,d4) with elements in {2,4,6}.

			The term 3299 corresponds to the quadruple (2,6,6,6): 3299, 3301, 3307, 3313, 3319 are consecutive primes.

The quadruples are in A078868. The same primes, in lexicographic order of the quadruples, are in A078866. The sequences of primes corresponding to the 26 difference patterns are in A022006, A022007 and A078946-A078970. Cf. A001223.

Edited by Dean Hickerson, Dec 20 2002

A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

Original entry on oeis.org

3301, 15901, 18211, 30091, 53611, 71341, 77551, 80911, 89101, 120811, 252151, 285451, 292471, 294781, 344251, 601801, 616501, 744811, 792691, 809821, 908521, 912391, 1152631, 1154221, 1279801, 1376491, 1398031, 1455361, 1464271, 1500511, 1503031, 1555111, 1594261

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+6, p+12, p+18 and p+22 are consecutive primes.

			30091 is in the sequence since 30091, 30097 = 30091 + 6, 30103 = 30091 + 12, 30109 = 30091 + 18 and 30113 = 30091 + 22 are consecutive primes.

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

Subsequence of A033451. - R. J. Mathar, May 06 2017

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

Select[Partition[Prime[Range[150000]], 5, 1], Differences[#] == {6,6,6,4} &][[;;, 1]] (* Amiram Eldar, Feb 22 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 6 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 4, 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

A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

Original entry on oeis.org

2424, 2462, 2466, 2642, 2646, 2664, 2666, 4242, 4246, 4264, 4624, 4626, 4662, 4666, 6246, 6264, 6266, 6424, 6426, 6462, 6466, 6626, 6642, 6646, 6662, 6664

Offset: 1

Labos Elemer, Dec 19 2002

			4624 corresponds to the quadruple (4,6,2,4). It is in the sequence because the 5 consecutive primes 1597, 1601, 1607, 1609 and 1613 have differences (4,6,2,4).

The least primes corresponding to the quadruples are in A078866. The same primes, in increasing order, are in A078867. The sequences of primes corresponding to the 26 difference patterns are in A022006 (for 2424), A022007 (for 4242) and A078946-A078970. The similarly defined quintuples are in A078870. Cf. A001223.

With[{k = 4}, FromDigits /@ Select[Tuples[Range[2, 6, 2], k], Function[m, Count[Range[k, 10^k], n_ /; Times @@ Boole@ Map[PrimeQ, Prime@ n + Accumulate@ m] == 1] > 0]]] (* Michael De Vlieger, Mar 25 2017 *) (* or *)
FromDigits /@ Union@ Select[ Partition[ Differences@ Prime@ Range[3, 2000], 4, 1], Max@ # <= 6 &] (* Giovanni Resta, Mar 25 2017 *)

Edited by Dean Hickerson, Dec 20 2002

A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

Original entry on oeis.org

41, 641, 1091, 4001, 9461, 26681, 26711, 44531, 79811, 103991, 110921, 112571, 172421, 223241, 276821, 289841, 290021, 317771, 373181, 381371, 434921, 450881, 493121, 602081, 678761, 788351, 834131, 907211, 974861, 1076501, 1081121, 1097891, 1200371, 1409531, 1426151

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+6, p+12 and p+18 are consecutive primes.

			641 is in the sequence since 641, 643 = 641 + 2, 647 = 641 + 6, 653 = 641 + 12 and 659 = 641 + 18 are consecutive primes.

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

Subsequence of A078847.

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 4, 6, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 11 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

Original entry on oeis.org

17, 1277, 1607, 3527, 4637, 71327, 97367, 113147, 191447, 290657, 312197, 416387, 418337, 421697, 450797, 566537, 795647, 886967, 922067, 1090877, 1179317, 1300127, 1464257, 1632467, 1749257, 1866857, 1901357, 2073347, 2322107

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

The last digit of each of these prime numbers is 7.
Subsequence of A078946.
The primes are always consecutive: The few ways of inserting other primes are: (p,p+2,p+4)... [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+8),(p+12),(p+14) [impossible since one of these would be a multiple of 5]; (p,p+2,p+6),(p+10) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+16) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+18) [impossible since one of these would be a multiple of 5]. - R. J. Mathar, Jun 15 2013

			The first two terms correspond to the sextuples (17,19,23,29,31,37) and (1277,1279,1283,1289,1291,1297).

R. K. Guy, Unsolved Problems in Number Theory, E30.

Amiram Eldar, Table of n, a(n) for n = 1..1000
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
T. Forbes, Prime k-tuplets
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
Eric Weisstein's World of Mathematics, Prime Triplet.

Initial members of prime quadruples (p, p+2, p+6, p+12): A172454.

Cf. A073648, A098412.

for n from 1 by 2 to 400000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) and isprime(n+20) then print(n) else fi;od;

Select[Prime[Range[171000]],And@@PrimeQ[{#+2,#+6,#+12,#+14,#+20}]&] (* Harvey P. Dale, Jul 23 2011 *)
Select[Prime[Range[171000]],AllTrue[#+{2,6,12,14,20},PrimeQ]&] (* or *) Select[ Partition[Prime[Range[171000]],6,1],Differences[#]=={2,4,6,2,6}&][[All,1]] (* Harvey P. Dale, Sep 04 2022 *)

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

A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

Original entry on oeis.org

29, 59, 269, 1289, 2129, 2789, 5639, 8999, 13679, 14549, 18119, 36779, 62129, 75989, 80669, 83219, 88799, 93479, 113159, 115769, 124769, 132749, 150209, 160079, 163979, 203309, 207509, 223829, 228509, 278489, 282089, 284729, 298679, 312929, 313979, 323369, 337859

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+14 are consecutive primes.

All terms are congruent to 29 (mod 30). - Muniru A Asiru, Sep 04 2017

			59 is in the sequence since 59, 61 = 59 + 2, 67 = 59 + 8, 71 = 59 + 12 and 73 = 59 + 14 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. J. Mathar)
R. J. Mathar, Table of Prime Gap Constellations.

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

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

K:=26*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);;  I:=[2,6,4,2];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
Q:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3]]),I),i->P[i]); # Muniru A Asiru, Sep 04 2017

for i from 1 to 10^5 do if [ithprime(i+1),ithprime(i+2),ithprime(i+3),ithprime(i+4)] = [ithprime(i)+2,ithprime(i)+8,ithprime(i)+12,ithprime(i)+14] then print(ithprime(i)); fi; od;  # Muniru A Asiru, Sep 04 2017

Select[Partition[Prime[Range[26000]],5,1],Differences[#]=={2,6,4,2}&][[;;,1]] (* Harvey P. Dale, Dec 10 2024 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A078949 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,6).

Original entry on oeis.org

71, 431, 2339, 2381, 5849, 6959, 27791, 32561, 41609, 45119, 46439, 48479, 51419, 54401, 63599, 78779, 81551, 106859, 115319, 130631, 138569, 143501, 153269, 166601, 183569, 196169, 204359, 229751, 246929, 266081, 279119, 321311, 326999, 350729, 357659, 362741

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+12 and p+18 are consecutive primes.

			71 is in the sequence since 71, 73 = 71 + 2, 79 = 71 + 8, 83 = 71 + 12 and 89 = 71 + 18 are consecutive primes.

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

Subsequence of A078848. - R. J. Mathar, Feb 10 2013

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 6, 4, 6} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 4 && p5 - p4 == 6, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

From Amiram Eldar, Feb 21 2025: (Start)
a(n) == 5 (mod 6).
a(n) == 11 or 29 (mod 30). (End)

Edited by Dean Hickerson, Dec 20 2002

A078950 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,6,4).

Original entry on oeis.org

149, 599, 27059, 31319, 42179, 65699, 75209, 85829, 87539, 92219, 135599, 170759, 205949, 221069, 249419, 274829, 278609, 280589, 287849, 302579, 307259, 308309, 350429, 355499, 398339, 406499, 416399, 422549, 541529, 566549, 573479, 585839, 603899, 609599, 637709

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+2, p+8, p+14 and p+18 are consecutive primes.

			149 is in the sequence since 149, 151 = 149 + 2, 157 = 149 + 8, 163 = 149 + 14 and 167 = 149 + 18 are consecutive primes.

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

Subsequence of A078849. - R. J. Mathar, May 06 2017

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

Select[Partition[Prime[Range[50000]], 5, 1], Differences[#] == {2, 6, 6, 4} &][[;;, 1]] (* Amiram Eldar, Feb 21 2025 *)

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 6 && p4 - p3 == 6 && p5 - p4 == 4, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 29 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

cp's OEIS Frontend

A078866 The quadruples (d1,d2,d3,d4) with elements in {2,4,6} are listed in lexicographic order; for each quadruple, this sequence lists the smallest prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4), if such a prime exists.

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A078867 Sorted version of A078866.

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A078969 Primes p such that the differences between the 5 consecutive primes starting with p are (6,6,6,4).

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A078868 Decimal concatenations of the quadruples (d1,d2,d3,d4) with elements in {2,4,6} for which there exists a prime p >= 5 such that the differences between the 5 consecutive primes starting with p are (d1,d2,d3,d4).

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A078947 Primes p such that the differences between the 5 consecutive primes starting with p are (2,4,6,6).

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A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

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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.)

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Programs

Mathematica

Extensions

A078948 Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).

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