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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

The array starts with:

[18713, 18719, 18731, 18743, 18749]

[25603, 25609, 25621, 25633, 25639]

[28051, 28057, 28069, 28081, 28087]

...

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.

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

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Last modified October 17 07:39 EDT 2021. Contains 348048 sequences. (Running on oeis4.)