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.
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
KEYWORD
nonn,tabf
AUTHOR
Michel Lagneau, Feb 23 2016
STATUS
approved