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A190939 Subgroups of nimber addition interpreted as binary numbers.

%I A190939 #64 Jun 10 2023 03:14:55
%S A190939 1,3,5,9,15,17,33,51,65,85,105,129,153,165,195,255,257,513,771,1025,
%T A190939 1285,1545,2049,2313,2565,3075,3855,4097,4369,4641,5185,6273,8193,
%U A190939 8481,8721,9345,10305,12291,13107,15555,16385,16705,17025,17425,18465,20485,21845
%N A190939 Subgroups of nimber addition interpreted as binary numbers.
%C A190939 Each subgroup {0,a,b,...} of nimber addition can be assigned an integer 1+2^a+2^b+...
%C A190939 These integers ordered by size give this sequence.
%C A190939 Without nimbers the sequence may be defined as follows:
%C A190939 The powerset af a set {0,...,n-1} with the symmetric difference as group operation forms the elementary abelian group (Z_2)^n.
%C A190939 The elements of the group can be numbered lexicographically from 0 to 2^n-1, with 0 representing the neutral element:
%C A190939 {}-->0 , {0}-->2^0=1 , {1}-->2^1=2 , {0,1}-->2^0+2^1=3 , ... , {0,...,n-1}-->2^n-1
%C A190939 So the subgroups of (Z_2)^n can be represented by subsets of {0,...,2^n-1}.
%C A190939 So each subgroup {0,a,b,...} of (Z_2)^n can be assigned an integer 1+2^a+2^b+...
%C A190939 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 A190939 This leads to the infinite sequence:
%C A190939 * 1,     (1 until here for (Z_2)^0)
%C A190939 * 3,     (2 until here for (Z_2)^1)
%C A190939 * 5, 9, 15,     (5 until here for (Z_2)^2)
%C A190939 * 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (16 until here for (Z_2)^3)
%C A190939 * 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 A190939 * 65537, ...
%C A190939 The number of subgroups of (Z_2)^n is 1, 2, 5, 16, 67, 374, 2825, ... (A006116)
%C A190939 Comment from _Tilman Piesk_, Aug 27 2013: (Start)
%C A190939 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 A190939 (End)
%H A190939 Tilman Piesk, <a href="/A190939/b190939.txt">Table of n, a(n) for n = 0..2824</a>
%H A190939 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 A190939 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 A190939 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Subgroups_of_nimber_addition">Subgroups of nimber addition</a> (Wikiversity)
%e A190939 The 5 subgroups of the Klein four-group (Z_2)^2 and corresponding integers are:
%e A190939 {0      }     -->     2^0                     =   1
%e A190939 {0,1    }     -->     2^0 + 2^1               =   3
%e A190939 {0,  2  }     -->     2^0       + 2^2         =   5
%e A190939 {0,    3}     -->     2^0             + 2^3   =   9
%e A190939 {0,1,2,3}     -->     2^0 + 2^1 + 2^2 + 2^3   =  15
%Y A190939 Cf. A227963 (the same small equivalence classes represented by entries of A227722)
%Y A190939 Cf. A198260 (number of runs of ones in the binary strings)
%Y A190939 Subsequences:
%Y A190939 Cf. A051179 (2^2^n-1).
%Y A190939 Cf. A083318 (2^n+1).
%Y A190939 Cf. A001317 (rows of the Sierpinski triangle read like binary numbers).
%Y A190939 Cf. A228540 (rows of negated binary Walsh matrices r.l.b.n.).
%Y A190939 Cf. A122569 (negated iterations of the Thue-Morse sequence r.l.b.n.).
%K A190939 nonn,tabf
%O A190939 0,2
%A A190939 _Tilman Piesk_, May 24 2011
%E A190939 Offset changed to 0 by _Tilman Piesk_, Jan 25 2012