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

Offset: 0

Author

Tilman Piesk, Jul 22 2013

Keywords

• nonn

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.

Examples

			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)
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
		

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

Crossrefs

Cf. A227723 (subsequence that does the same thing for big equivalence classes).

Cf. A000231, A051179, A000215.

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