A190939 - OEIS

%I #64 Jun 10 2023 03:14:55

%S 1,3,5,9,15,17,33,51,65,85,105,129,153,165,195,255,257,513,771,1025,

%T 1285,1545,2049,2313,2565,3075,3855,4097,4369,4641,5185,6273,8193,

%U 8481,8721,9345,10305,12291,13107,15555,16385,16705,17025,17425,18465,20485,21845

%N Subgroups of nimber addition interpreted as binary numbers.

%C Each subgroup {0,a,b,...} of nimber addition can be assigned an integer 1+2^a+2^b+...

%C These integers ordered by size give this sequence.

%C Without nimbers the sequence may be defined as follows:

%C The powerset af a set {0,...,n-1} with the symmetric difference as group operation forms the elementary abelian group (Z_2)^n.

%C The elements of the group can be numbered lexicographically from 0 to 2^n-1, with 0 representing the neutral element:

%C {}-->0 , {0}-->2^0=1 , {1}-->2^1=2 , {0,1}-->2^0+2^1=3 , ... , {0,...,n-1}-->2^n-1

%C So the subgroups of (Z_2)^n can be represented by subsets of {0,...,2^n-1}.

%C So each subgroup {0,a,b,...} of (Z_2)^n can be assigned an integer 1+2^a+2^b+...

%C For each (Z_2)^n there is a finite sequence of these numbers ordered by size, and it is the beginning of the finite sequence for (Z_2)^(n+1).

%C This leads to the infinite sequence:

%C * 1,     (1 until here for (Z_2)^0)

%C * 3,     (2 until here for (Z_2)^1)

%C * 5, 9, 15,     (5 until here for (Z_2)^2)

%C * 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (16 until here for (Z_2)^3)

%C * 257, 513, 771, 1025, 1285, 1545, 2049, 2313, 2565, 3075, 3855, 4097, 4369, 4641, 5185, 6273, 8193, 8481, 8721, 9345, 10305, 12291, 13107, 15555, 16385, 16705, 17025, 17425, 18465, 20485, 21845, 23205, 24585, 26265, 26985, 32769, 33153, 33345, 33825, 34833, 36873, 38505, 39321, 40965, 42405, 43605, 49155, 50115, 52275, 61455, 65535,     (67 until here for (Z_2)^4)

%C * 65537, ...

%C The number of subgroups of (Z_2)^n is 1, 2, 5, 16, 67, 374, 2825, ... (A006116)

%C Comment from _Tilman Piesk_, Aug 27 2013: (Start)

%C Boolean functions correspond to integers, and belong to small equivalence classes (sec). So a sec can be seen as an infinite set of integers (represented in A227722 by the smallest one). Some secs contain only one odd integer. These unique odd integers, ordered by size, are shown in this sequence. (While the smallest integers from these secs are shown in A227963.)

%C (End)

%H Tilman Piesk, <a href="/A190939/b190939.txt">Table of n, a(n) for n = 0..2824</a>

%H Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Z2%5E4;_subgroups_list.svg#File">Graphical explanation of n, a(n), A227963(n) for n = 0..66</a>

%H Tilman Piesk, <a href="/A190939/a190939.txt">2825x64 submatrix</a> of the corresponding binary array, <a href="/A190939/a190939_1.txt">corresponding Walsh spectra</a> (<a href="http://en.wikiversity.org/wiki/Subgroups_of_Z2%5E6">human readable versions</a> of these matrices)

%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Subgroups_of_nimber_addition">Subgroups of nimber addition</a> (Wikiversity)

%e The 5 subgroups of the Klein four-group (Z_2)^2 and corresponding integers are:

%e {0      }     -->     2^0                     =   1

%e {0,1    }     -->     2^0 + 2^1               =   3

%e {0,  2  }     -->     2^0       + 2^2         =   5

%e {0,    3}     -->     2^0             + 2^3   =   9

%e {0,1,2,3}     -->     2^0 + 2^1 + 2^2 + 2^3   =  15

%Y Cf. A227963 (the same small equivalence classes represented by entries of A227722)

%Y Cf. A198260 (number of runs of ones in the binary strings)

%Y Subsequences:

%Y Cf. A051179 (2^2^n-1).

%Y Cf. A083318 (2^n+1).

%Y Cf. A001317 (rows of the Sierpinski triangle read like binary numbers).

%Y Cf. A228540 (rows of negated binary Walsh matrices r.l.b.n.).

%Y Cf. A122569 (negated iterations of the Thue-Morse sequence r.l.b.n.).

%K nonn,tabf

%O 0,2

%A _Tilman Piesk_, May 24 2011

%E Offset changed to 0 by _Tilman Piesk_, Jan 25 2012