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Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.
5

%I #37 Apr 02 2024 14:33:52

%S 3,5,7,9,13,15,17,25,29,31,33,49,57,61,63,65,97,113,121,125,127,129,

%T 193,225,241,249,253,255,257,385,449,481,497,505,509,511,513,769,897,

%U 961,993,1009,1017,1021,1023,1025,1537,1793,1921,1985,2017,2033,2041,2045,2047

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

%C 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. Using the lexicographic ordering of A057555 the sequence is:

%C A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...

%C +1 | 2 4 8 16 32 64 128 256 512 1024 ...

%C ----|-----------------------------------------------------------------

%C 1 | 3 5 9 17 33 65 129 257 513 1025

%C 3 | 7 13 25 49 97 193 385 769 1537 3073

%C 7 | 15 29 57 113 225 449 897 1793 3585 7169

%C 15 | 31 61 121 241 481 961 1921 3841 7681 15361

%C 31 | 63 125 249 497 993 1985 3969 7937 15873 31745

%C 63 | 127 253 505 1009 2017 4033 8065 16129 32257 64513

%C 127 | 255 509 1017 2033 4065 8129 16257 32513 65025 130049

%C 255 | 511 1021 2041 4081 8161 16321 32641 65281 130561 261121

%C 511 | 1023 2045 4089 8177 16353 32705 65409 130817 261633 523265

%C 1023| 2047 4093 8185 16369 32737 65473 130945 261889 523777 1047553

%C ...

%C All of these numbers have the following property:

%C let m be a member of A(n),

%C if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then

%C the differences between successive members of B(n) is a repeating series

%C of 1's with the last difference in the pattern m. The number of ones in

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

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

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

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

%C with successive differences of:

%C 1, 1, 1, 1, 1, 1, 1, 9.

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

%C The formula for diagonals above the main diagonal

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

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

%C The formulas for diagonals below the main diagonal

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

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

%C Primes of this sequence are in A152449.

%H Brad Clardy, <a href="/A224195/b224195.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (2^(A057555(2*n) - 1)*2^(A057555(2*n-1)-A057555(2*n)) + 1 for n>=1.

%F 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

%t Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* _Paolo Xausa_, Apr 02 2024 *)

%o (Magma)

%o //program generates values in a table form

%o for i:=1 to 10 do

%o m:=2^i - 1;

%o m,[ m*2^n +1 : n in [1..10]];

%o end for;

%o //program generates sequence in lexicographic ordering of A057555, read

%o //along antidiagonals from top. Primes in the sequence are marked with *.

%o for i:=2 to 18 do

%o for j:=1 to i-1 do

%o m:=2^j -1;

%o k:=m*2^(i-j) + 1;

%o if IsPrime(k) then k,"*";

%o else k;

%o end if;;

%o end for;

%o end for;

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

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

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

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

%K nonn,tabl

%O 1,1

%A _Brad Clardy_, Apr 01 2013