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

FORMULA

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 2) + 3 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 +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;

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.

Sequence in context: A129916 A032694 A004769 * A125771 A158290 A328896

Adjacent sequences:  A224377 A224378 A224379 * A224381 A224382 A224383

KEYWORD

tabl,nonn

AUTHOR

Brad Clardy, Apr 05 2013

STATUS

approved

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Last modified April 20 11:33 EDT 2021. Contains 343135 sequences. (Running on oeis4.)