A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Author

Brad Clardy, Apr 01 2013

Keywords

• nonn
• tabl

Comments

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. 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.

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's with the last difference in the pattern m. The number of ones in

the pattern is 2^j - 1, where j is the column index.

As an example consider A(4) which is 9,

the sequence B(n) where i XOR 8 = i - 8 starts as:

8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)

with successive differences of:

1, 1, 1, 1, 1, 1, 1, 9.

The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).

The formula for diagonals above the main diagonal

2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal

2^(2*n) - 2^(n + (b/2)) + 1 n>=(b/2)+1 b=even number above diagonal

The formulas for diagonals below the main diagonal

2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal

2^(2*n) - 2^(n - (b/2)) + 1 n>=(b/2)+1 b=even number below diagonal

Primes of this sequence are in A152449.

Examples

			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)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...
		

Links

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

Crossrefs

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).

Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).

Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).

Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

Programs

Magma

//program generates values in a table form
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^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Mathematica

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

Formula

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013