A267028 P(n,k) is an array read by rows, with n > 0 and k=1..5, where row n gives the chain of 5 consecutive primes {p(i), p(i+1), p(i+2), p(i+3), p(i+4)} having the symmetrical property p(i) + p(i+4) = p(i+1) + p(i+3) = 2*p(i+2) for some index i.

18713, 18719, 18731, 18743, 18749, 25603, 25609, 25621, 25633, 25639, 28051, 28057, 28069, 28081, 28087, 30029, 30047, 30059, 30071, 30089, 31033, 31039, 31051, 31063, 31069, 44711, 44729, 44741, 44753, 44771, 76883, 76907, 76913, 76919, 76943

Offset

1,1

Comments

a(3 + 5*(n-1)) = A051795(n).

The immediate objective of the sequence is to examine symmetrical properties in the array P(n,k). It is interesting to note that the results with the dimension 5 are generalizable to the dimensions 7, 9, ...

Notation:

We introduce the following function S(i,j) where row i is defined by {P(i,k)} and row j is defined by {P(j,k)}, k = 1..5. Let S(i, j) = 1 if P(i,1) + P(j,5) = P(i,2) + P(j,4) = P(i,3) + P(j,3), otherwise 0.

Conjecture:

For each integer n, there exists an infinite sequence of integers b(n,m), m = 1, 2, ... such that S(n, b(n,m)) = 1.

The following table gives the first values b(n,m).

Notation in the table: "PS" = primitive sequence.

+----+------------------------------------------------+-----------+

| n | sequences b(n,m), m=1,2,... of index |included in|

+----+------------------------------------------------+-----------+

| 1 | 1, 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, ... | PS |

| 2 | 2, 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ...| {b(1,m)} |

| 3 | 3, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, ... | {b(1,m)} |

| 4 | 4, 6, 11, 13, 14, 21, 28, 35, 39, 57, 59, ... | PS |

| 5 | 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, ... | {b(1,m)} |

| 6 | 6, 11, 13, 14, 21, 35, 39, 57, 59, 63, 67, ...| {b(4,m)} |

| 7 | 7, 30, 52, 55, 73, 74, 115, 159, 177, 183, ...| PS |

| 8 | 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 23, ... | {b(1,m)} |

| 9 | 9, 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, ...| {b(1,m)} |

| 10 | 10, 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, ...| {b(1,m)} |

| 11 | 11, 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, ...| {b(4,m)} |

| 12 | 12, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, ...| {b(1,m)} |

| 13 | 13, 14, 21, 28, 35, 39, 57, 59, 63, 67, 70, ...| {b(4,m)} |

| .. | ... | ... |

| 20 | 20, 43, 56, 96, 113, 131, 135, 156, 196, ... | PS |

| 25 | 21, 33, 37, 38, 40, 47, 48, 65, 76, 79, 83, ...| PS |

...

Example: S(7, 30) = 1.

We observe primitive sequences {b(n,m)} for n = {1, 4, 7, 20, 25, ...}.

(A primitive sequence is a sequence which is not included in another.)

Properties:

(1) S(i, i)= 1 for all i;

(2) S(i, j) = 1 => S(j, i) = 1;

(3) S(i, j) = 1 and S(j, L) = 1 => S(i, L) = 1.

Example:

For n = 1, {P(1,k)} = {18713, 18719, 18731, 18743, 18749};

we choose, for instance, b(1,2) = 3 => for n = 3, {C(3,k)} = {28051, 28057, 28069, 28081, 28087};

S(1,3) = 1 because 18713 + 28087 = 18719 + 28081 = 18731 + 28069 = 18743 + 28057 = 18749 + 28051 = 46800.

In order to find the index L for satisfying the property (3), we choose, for instance, the index b(3,2) = 8 => for n = 8, {P(8,k)} = {97423, 97429, 97441, 97453, 97459} and S(3, 8) = 1 because 28051 + 97459 = 28057 + 97453 = 28069 + 97441 = 28081 + 97429 = 28087 + 97423 = 125510.

Conclusion: S(1, 3) = 1 and S(3, 8) = 1 => S(1, 8) = 1 with 18713 + 97459 = 18719 + 97453 = 18731 + 97441 = 18743 + 97429 = 18749 + 97423 = 116172.

Links

Table of n, a(n) for n=1..35.

Example

The first row is [18713, 18719, 18731, 18743, 18749] because 18713 + 18749 = 18719 + 18743 = 2*18731 = 37462.
The array starts with:
  [18713, 18719, 18731, 18743, 18749]
  [25603, 25609, 25621, 25633, 25639]
  [28051, 28057, 28069, 28081, 28087]
  ...

Maple

U:=array(1..50, 1..5):W:=array(1..2):kk:=0:
for n from 4 to 10000 do:
   for m from 2 by -1 to 1 do:
      q:=ithprime(n-m)+ithprime(n+m):W[m]:=q:
    od:
    if W[1]=W[2] and W[1]=2*ithprime(n) then
    kk:=kk+1:U[kk, 1]:=ithprime(n-2):
    U[kk, 2]:=ithprime(n-1):U[kk, 3]:=ithprime(n):
    U[kk, 4]:=ithprime(n+1):U[kk, 5]:=ithprime(n+2):
    else fi:od:print(U):
    for i from 1 to kk do:
     for j from i+1 to kk do:
      s1:=U[i, 1]+U[j, 5]:
      s2:=U[i, 2]+U[j, 4]:
      s3:=U[i, 3]+U[j, 3]:
      s4:=U[i, 4]+U[j, 2]:
      s5:=U[i, 5]+U[j, 1]:
     if s1=s2 and s2=s3 and s3=s4 and s4=s5
     then
     printf("%d %d \n", i, j):
     else fi:
     od:
  od:

Crossrefs

Cf. A051795, A055380, A055382.

Sequence in context: A248065 A283934 A266061 * A226150 A081416 A051795

Adjacent sequences: A267025 A267026 A267027 * A267029 A267030 A267031

Keyword

nonn,tabf

Author

Michel Lagneau, Feb 23 2016

Status

approved