

A227722


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


5



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, 55, 60, 61, 63, 85, 86, 87, 90, 91, 95, 102, 103, 105, 107, 111, 119, 123, 125, 126, 127, 255, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

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 2colored vertices that are equivalent up to reflection (i.e., exchanging opposite hyperfaces).
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 nary 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.


LINKS

Tilman Piesk, Table of n, a(n) for n = 0..9999
Tilman Piesk, Small equivalence classes of Boolean functions
Tilman Piesk, sec of 3ary functions corresponding to a(12) = 22 = 0x16
Tilman Piesk, MATLAB code used for the calculation
Index entries for sequences related to Boolean functions


FORMULA

a( A000231  1 ) = a(2,6,45,4335...) = 3,15,255,65535... = A051179
a( A000231 ) = a(3,7,46,4336...) = 5,17,257,65537... = A000215


EXAMPLE

The 16 2ary functions ordered in A000231(2) = 7 small equivalence classes:
a a(n) Boolean functions, the left one corresponding to a(n)
0 0 0000
1 1 0001, 0010, 0100, 1000
2 3 0011, 1100
3 5 0101, 1010
4 6 0110, 1001
5 7 0111, 1011, 1101, 1110
6 15 1111


CROSSREFS

Cf. A227723 (subsequence that does the same thing for big equivalence classes).
Cf. A000231, A051179, A000215.
Sequence in context: A281725 A274928 A163620 * A250419 A072134 A179220
Adjacent sequences: A227719 A227720 A227721 * A227723 A227724 A227725


KEYWORD

nonn


AUTHOR

Tilman Piesk, Jul 22 2013


STATUS

approved



