A227722 - OEIS

A227722

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

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 2-colored 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 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.

LINKS

Tilman Piesk, Table of n, a(n) for n = 0..9999

Tilman Piesk, Small equivalence classes of Boolean functions

Tilman Piesk, sec of 3-ary 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 2-ary functions ordered in A000231(2) = 7 small equivalence classes:

a a(n) Boolean functions, the left one corresponding to a(n)