A078969 -id:A078969 - cp's OEIS frontend

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.)

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

A078955 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,2,6).

Original entry on oeis.org

19, 1279, 1609, 2539, 3529, 4639, 5839, 15259, 19069, 32359, 71329, 75979, 88789, 97369, 112909, 113149, 130639, 135589, 138559, 191449, 229759, 246919, 290659, 312199, 346429, 349369, 357649, 384469, 396619, 416389, 418339, 421699, 433249, 435559, 450799, 460969

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+10, p+12 and p+18 are consecutive primes.

			19 is in the sequence since 19, 23 = 19 + 4, 29 = 19 + 10, 31 = 19 + 12 and 37 = 19 + 18 are consecutive primes.

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

Subsequence of A078851. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078969, A022006, A022007.

Transpose[Select[Partition[Prime[Range[40000]],5,1],Differences[#]=={4,6,2,6}&]][[1]]  (* Harvey P. Dale, Feb 03 2011 *)

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

a(n) == 19 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A078956 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,2).

Original entry on oeis.org

43, 163, 643, 1213, 2953, 4003, 7573, 11923, 14533, 25453, 26683, 26713, 29863, 41593, 48523, 61543, 68473, 150193, 151153, 172423, 206803, 227593, 290023, 302563, 338563, 343813, 346543, 428023, 527053, 529033, 540373, 547483, 551713, 570403, 577513, 622603, 628993

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+10, p+16 and p+18 are consecutive primes.

All terms == 13 (mod 30). - Robert Israel, Oct 17 2023

			43 is in the sequence since 43, 47 = 43 + 4, 53 = 43 + 10, 59 = 43 + 16 and 61 = 43 + 18 are consecutive primes.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert Israel)

Subsequence of A078852. - R. J. Mathar, May 06 2017

Cf. A001223, A078866, A078867, A078946-A078969, A022006, A022007.

L:= [0$5]:
p:= 1: R:= NULL: count:= 0:
while count < 100 do
  p:= nextprime(p);
  L:= [L[2],L[3],L[4],L[5],p];
  if L -~ L[1] = [0, 4, 10, 16, 18] then
    count:= count+1;
    R:= R, L[1];
  fi
od:
R; # Robert Israel, Oct 17 2023

Select[Partition[Prime[Range[50000]],5,1],Differences[#]=={4,6,6,2}&][[All,1]] (* Harvey P. Dale, Jan 23 2021 *)

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

A078957 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,6).

Original entry on oeis.org

12637, 14737, 15787, 17467, 78787, 95257, 104707, 120997, 154057, 243517, 250027, 252877, 351037, 357667, 443227, 496477, 501187, 593497, 624787, 696607, 750787, 917827, 949957, 1003087, 1025257, 1104097, 1109887, 1260877, 1279657, 1457857, 1517917, 1565167, 1654717

Offset: 1

Labos Elemer, Dec 19 2002

nonn

Equivalently, primes p such that p, p+4, p+10, p+16 and p+22 are consecutive primes.

			15787 is in the sequence since 15787, 15791 = 15787 + 4, 15797 = 15787 + 10, 15803 = 15787 + 16 and 15809 = 15787 + 22 are consecutive primes.

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

Cf. A001223, A078866, A078867, A078946-A078969, A022006, A022007.

Select[Partition[Prime[Range[10^5]],5,1],Differences[#]=={4,6,6,6}&][[All,1]] (* Harvey P. Dale, Jun 23 2019 *)

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

a(n) == 7 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A079021 Suppose p and q = p+22 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 51 possible difference patterns, shown in the Comments line. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 241, 271, 331, 337, 397, 409, 421, 457, 487, 499, 541, 619, 661, 739, 751, 787, 919, 991, 1069, 1129, 1471, 1531, 1597, 1867, 2221, 2287, 2671, 2707, 2797, 2857, 3187, 3301, 3391, 3637, 4651, 6547, 12637, 17011, 90001

Offset: 1

Labos Elemer, Jan 24 2003

The 51 difference patterns are [22], [4,18], [6,16], [10,12], [12,10], [16,6], [18,4], [4,2,16], [4,6,12], [4,8,10], [4,12,6], [4,14,4], [6,4,12], [6,6,10], [6,10,6], [6,12,4], [10,2,10], [10,6,6], [10,8,4], [12,4,6], [12,6,4], [16,2,4], [4,2,4,12], [4,2,6,10], [4,2,10,6], [4,6,2,10], [4,6,6,6], [4,6,8,4], [4,8,4,6], [4,8,6,4], [6,4,2,10], [6,4,6,6], [6,4,8,4], [6,6,4,6], [6,6,6,4], [6,10,2,4], [10,2,4,6], [10,2,6,4], [10,6,2,4], [12,4,2,4], [4,2,4,2,10], [4,2,4,6,6], [4,2,6,4,6], [4,6,2,4,6], [4,6,2,6,4], [6,4,2,4,6], [6,4,2,6,4], [6,4,6,2,4], [6,6,4,2,4], [10,2,4,2,4], [4,2,4,2,4,6].

Certain patterns are singular, i.e. occur only once like [4,2,4,2,4,6].

			p=6547, q=6569 has difference pattern [4,2,10,6] and {6547,6551,6553,6563,6569} is the corresponding consecutive prime 5-tuple.

A078957(1)=12637, A078964(1)=157, A078967(1)=151, A078969(1)=3301, A000230(11)=1129. Cf. A079016-A079024.

cp's OEIS Frontend

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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A078955 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,2,6).

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A078956 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,2).

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A078957 Primes p such that the differences between the 5 consecutive primes starting with p are (4,6,6,6).

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