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A227722 Smallest Boolean functions from small equivalence classes (counted by A000231).

%I A227722 #29 Dec 16 2017 22:43:34
%S A227722 0,1,3,5,6,7,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,51,53,54,
%T A227722 55,60,61,63,85,86,87,90,91,95,102,103,105,107,111,119,123,125,126,
%U A227722 127,255,257,258,259,260,261,262,263,264,265,266,267
%N A227722 Smallest Boolean functions from small equivalence classes (counted by A000231).
%C A227722 Two Boolean functions belong to the same small equivalence class (sec) when they can be expressed by each other by negating arguments. E.g., when f(p,~q,r) = g(p,q,r), then f and g belong to the same sec. Geometrically this means that the functions correspond to hypercubes with 2-colored vertices that are equivalent up to reflection (i.e., exchanging opposite hyperfaces).
%C A227722 Boolean functions correspond to integers, so each sec can be denoted by the smallest integer corresponding to one of its functions. There are A000231(n) small equivalence classes of n-ary Boolean functions. Ordered by size they form the finite sequence A_n. It is the beginning of A_(n+1) which leads to this infinite sequence A.
%H A227722 Tilman Piesk, <a href="/A227722/b227722.txt">Table of n, a(n) for n = 0..9999</a>
%H A227722 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Equivalence_classes_of_Boolean_functions#sec">Small equivalence classes of Boolean functions</a>
%H A227722 Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Boolean_functions_like_0110_1000.svg">sec of 3-ary functions</a> corresponding to a(12) = 22 = 0x16
%H A227722 Tilman Piesk, <a href="http://pastebin.com/kdZBTYnU">MATLAB code used for the calculation</a>
%H A227722 <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F A227722 a( A000231 - 1 ) = a(2,6,45,4335...) = 3,15,255,65535... = A051179
%F A227722 a( A000231 )     = a(3,7,46,4336...) = 5,17,257,65537... = A000215
%e A227722 The 16 2-ary functions ordered in A000231(2) = 7 small equivalence classes:
%e A227722 a     a(n)    Boolean functions, the left one corresponding to a(n)
%e A227722 0      0      0000
%e A227722 1      1      0001, 0010, 0100, 1000
%e A227722 2      3      0011, 1100
%e A227722 3      5      0101, 1010
%e A227722 4      6      0110, 1001
%e A227722 5      7      0111, 1011, 1101, 1110
%e A227722 6     15      1111
%Y A227722 Cf. A227723 (subsequence that does the same thing for big equivalence classes).
%Y A227722 Cf. A000231, A051179, A000215.
%K A227722 nonn
%O A227722 0,3
%A A227722 _Tilman Piesk_, Jul 22 2013