OFFSET
1,1
COMMENTS
The table has row labels 2^n - 1 and column labels 2^(m+3). The table entry is row*col + 5. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the anti-diagonals starting from the top left corner. Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+5 | 16 32 64 128 256 512 1024 ...
----|-------------------------------------------
1 | 21 37 69 133 261 517 1029
3 | 53 101 197 389 773 1541 3077
7 | 117 229 453 901 1797 3589 7173
15 | 245 485 965 1925 3845 7685 15365
31 | 501 997 1989 3973 7941 15877 31749
63 | 1013 2021 4037 8069 16133 32261 64517
127 | 2037 4069 8133 16261 32517 65029 130053
...
All of these numbers have the following property:
let m be a member of A(n),
if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
the differences between successive members of B(n) is a repeating series
of 1,1,1,5 ending with 1,1,1 and the last difference in the pattern m.
The total number of 1's and 5's in the pattern is 2^(j+2) - 1, where
j is the column index.
As an example consider A(1) which is 21,
the sequence B(n) where i XOR 20 = i - 20 starts as:
20, 21, 22, 23, 28, 29, 30, 31, 52, ...
with successive differences of:
1, 1, 1, 5, 1, 1, 1, 21.
for A(2) which is 37,
the sequence B(n) where i XOR 36 = i - 36 starts as:
36, 37, 38, 39, 44, 45, 46, 47, 52, 53, 54, 55, 60, 61, 62, 63, 100,...
with successive differences of:
1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 37
LINKS
Brad Clardy, Table of n, a(n) for n = 1..1000
PROG
(MAGMA)
//program generates values in a table form, row labels of 2^i -1
for i:=1 to 10 do
m:=2^i - 1;
m, [ m*2^(n+3) +5 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along anti-diagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
for j:=1 to i-1 do
m:=2^j -1;
k:=m*2^(3+i-j) + 5;
if IsPrime(k) then k, "*";
else k;
end if;
end for;
end for;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Brad Clardy, Apr 16 2013
STATUS
approved