A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1.

21, 37, 53, 69, 101, 117, 133, 197, 229, 245, 261, 389, 453, 485, 501, 517, 773, 901, 965, 997, 1013, 1029, 1541, 1797, 1925, 1989, 2021, 2037, 2053, 3077, 3589, 3845, 3973, 4037, 4069, 4085, 4101, 6149, 7173, 7685, 7941, 8069, 8133, 8165, 8181, 8197, 12293, 14341, 15365, 15877, 16133

Offset: 1

Author

Brad Clardy, Apr 16 2013

Keywords

• nonn
• tabl

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 antidiagonals 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

Crossrefs

Cf. A057555 (lexicographic ordering).

Rows: A168614(i=1), n>=4.

Cols: A220087(j=2), n>=6.

Programs

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 antidiagonals 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;

Formula

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 3) + 5 for n>=1.