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A224701 Table read by antidiagonals of numbers of form (2^n - 1)*2^(m+3) + 5 where n>=1, m>=1. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 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

FORMULA

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

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 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;

CROSSREFS

Cf. A057555 (lexicographic ordering).

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

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

Sequence in context: A043768 A083567 A109211 * A050782 A061906 A139768

Adjacent sequences:  A224698 A224699 A224700 * A224702 A224703 A224704

KEYWORD

nonn,tabl

AUTHOR

Brad Clardy, Apr 16 2013

STATUS

approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)