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

%I A224195 #47 Feb 23 2025 04:49:33
%S A224195 3,5,7,9,13,15,17,25,29,31,33,49,57,61,63,65,97,113,121,125,127,129,
%T A224195 193,225,241,249,253,255,257,385,449,481,497,505,509,511,513,769,897,
%U A224195 961,993,1009,1017,1021,1023,1025,1537,1793,1921,1985,2017,2033,2041,2045,2047
%N A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.
%C A224195 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.
%C A224195 All of these numbers have the following property:
%C A224195    let m be a member of A(n),
%C A224195    if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
%C A224195    the differences between successive members of B(n) is a repeating series
%C A224195    of 1's with the last difference in the pattern m. The number of ones in
%C A224195    the pattern is 2^j - 1, where j is the column index.
%C A224195 As an example consider A(4) which is 9,
%C A224195    the sequence B(n) where i XOR 8 = i - 8 starts as:
%C A224195    8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
%C A224195    with successive differences of:
%C A224195    1, 1, 1, 1, 1, 1, 1, 9.
%C A224195 The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
%C A224195 The formula for diagonals above the main diagonal
%C A224195   2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
%C A224195   2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
%C A224195 The formulas for diagonals below the main diagonal
%C A224195   2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
%C A224195   2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
%C A224195 Primes of this sequence are in A152449.
%H A224195 Brad Clardy, <a href="/A224195/b224195.txt">Table of n, a(n) for n = 1..1000</a>
%F A224195 a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by _Jason Yuen_, Feb 22 2025]
%F A224195 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
%e A224195 Using the lexicographic ordering of A057555 the sequence is:
%e A224195 A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
%e A224195   +1  |    2    4     8    16    32     64    128    256     512    1024 ...
%e A224195   ----|-----------------------------------------------------------------
%e A224195   1   |    3    5     9    17    33     65    129    257     513    1025
%e A224195   3   |    7   13    25    49    97    193    385    769    1537    3073
%e A224195   7   |   15   29    57   113   225    449    897   1793    3585    7169
%e A224195   15  |   31   61   121   241   481    961   1921   3841    7681   15361
%e A224195   31  |   63  125   249   497   993   1985   3969   7937   15873   31745
%e A224195   63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
%e A224195   127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
%e A224195   255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
%e A224195   511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
%e A224195   1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
%e A224195   ...
%t A224195 Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* _Paolo Xausa_, Apr 02 2024 *)
%o A224195 (Magma)
%o A224195 //program generates values in a table form
%o A224195 for i:=1 to 10 do
%o A224195     m:=2^i - 1;
%o A224195     m,[ m*2^n +1 : n in [1..10]];
%o A224195 end for;
%o A224195 //program generates sequence in lexicographic ordering of A057555, read
%o A224195 //along antidiagonals from top. Primes in the sequence are marked with *.
%o A224195 for i:=2 to 18 do
%o A224195     for j:=1 to i-1 do
%o A224195        m:=2^j -1;
%o A224195        k:=m*2^(i-j) + 1;
%o A224195        if IsPrime(k) then k,"*";
%o A224195           else k;
%o A224195        end if;;
%o A224195     end for;
%o A224195 end for;
%Y A224195 Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
%Y A224195 Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
%Y A224195 Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
%Y A224195 Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).
%K A224195 nonn,tabl
%O A224195 1,1
%A A224195 _Brad Clardy_, Apr 01 2013