A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

11, 19, 27, 35, 51, 59, 67, 99, 115, 123, 131, 195, 227, 243, 251, 259, 387, 451, 483, 499, 507, 515, 771, 899, 963, 995, 1011, 1019, 1027, 1539, 1795, 1923, 1987, 2019, 2035, 2043, 2051, 3075, 3587, 3843, 3971, 4035, 4067, 4083, 4091, 4099, 6147, 7171, 7683, 7939, 8067, 8131, 8163, 8179

Offset: 1

Author

Brad Clardy, Apr 05 2013

Keywords

• tabl
• nonn

Comments

The table has row labels 2^n - 1 and column labels 2^(m+2). The table entry is row*col + 3. 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)...

+3 | 8 16 32 64 128 256 512 ...

----|-------------------------------------------

1 | 11 19 35 67 131 259 515

3 | 27 51 99 195 387 771 1539

7 | 59 115 227 451 899 1795 3587

15 | 123 243 483 963 1923 3843 7683

31 | 251 499 995 1987 3971 7939 15875

63 | 507 1011 2019 4035 8067 16131 32259

127 | 1019 2035 4067 8131 16259 32515 65027

...

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 an alternating series of 1's and 3's with the last difference in the pattern m. The number of alternating 1's and 3's in the pattern is 2^(j+1) - 1, where j is the column index.

As an example consider A(1) which is 11, the sequence B(n) where i XOR 10 = i - 10 starts as 10, 11, 14, 15, 26, 27, 30, 31, 42, ... (A214864) with successive differences of 1, 3, 1, 11.

Main diagonal is A191341, the largest k such that k-1 and k+1 in binary representation have the same number of 1's and 0's

Links

• Brad Clardy, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A057555(lexicographic ordering), A214864(example), A224195.

Rows: A062729(i=1), A147595(2 n>=5), A164285(3 n>=3).

Cols: A168616(j=1 n>=4).

Diagonal: A191341.

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 +1 : 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^(2+i-j) + 3;
       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) + 2) + 3 for n>=1.