A227963 -id:A227963 - cp's OEIS frontend

A190939 Subgroups of nimber addition interpreted as binary numbers.

1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, 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

Offset: 0

Tilman Piesk, May 24 2011

Each subgroup {0,a,b,...} of nimber addition can be assigned an integer 1+2^a+2^b+...
These integers ordered by size give this sequence.
Without nimbers the sequence may be defined as follows:
The powerset af a set {0,...,n-1} with the symmetric difference as group operation forms the elementary abelian group (Z_2)^n.
The elements of the group can be numbered lexicographically from 0 to 2^n-1, with 0 representing the neutral element:
{}-->0 , {0}-->2^0=1 , {1}-->2^1=2 , {0,1}-->2^0+2^1=3 , ... , {0,...,n-1}-->2^n-1
So the subgroups of (Z_2)^n can be represented by subsets of {0,...,2^n-1}.
So each subgroup {0,a,b,...} of (Z_2)^n can be assigned an integer 1+2^a+2^b+...
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).
This leads to the infinite sequence:
* 1,     (1 until here for (Z_2)^0)
* 3,     (2 until here for (Z_2)^1)
* 5, 9, 15,     (5 until here for (Z_2)^2)
* 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255, (16 until here for (Z_2)^3)
* 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)
* 65537, ...
The number of subgroups of (Z_2)^n is 1, 2, 5, 16, 67, 374, 2825, ... (A006116)
Comment from Tilman Piesk, Aug 27 2013: (Start)
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.)
(End)

			The 5 subgroups of the Klein four-group (Z_2)^2 and corresponding integers are:
{0      }     -->     2^0                     =   1
{0,1    }     -->     2^0 + 2^1               =   3
{0,  2  }     -->     2^0       + 2^2         =   5
{0,    3}     -->     2^0             + 2^3   =   9
{0,1,2,3}     -->     2^0 + 2^1 + 2^2 + 2^3   =  15

Tilman Piesk, Table of n, a(n) for n = 0..2824
Tilman Piesk, Graphical explanation of n, a(n), A227963(n) for n = 0..66
Tilman Piesk, 2825x64 submatrix of the corresponding binary array, corresponding Walsh spectra (human readable versions of these matrices)
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Cf. A227963 (the same small equivalence classes represented by entries of A227722)
Cf. A198260 (number of runs of ones in the binary strings)
Subsequences:
Cf. A051179 (2^2^n-1).
Cf. A083318 (2^n+1).
Cf. A001317 (rows of the Sierpinski triangle read like binary numbers).
Cf. A228540 (rows of negated binary Walsh matrices r.l.b.n.).
Cf. A122569 (negated iterations of the Thue-Morse sequence r.l.b.n.).

Offset changed to 0 by Tilman Piesk, Jan 25 2012

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

cp's OEIS Frontend

A190939 Subgroups of nimber addition interpreted as binary numbers.

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