A227723 Smallest Boolean functions from big equivalence classes (counted by A000616).

0, 1, 3, 6, 7, 15, 22, 23, 24, 25, 27, 30, 31, 60, 61, 63, 105, 107, 111, 126, 127, 255, 278, 279, 280, 281, 282, 283, 286, 287, 300, 301, 303, 316, 317, 318, 319, 360, 361, 362, 363, 366, 367, 382, 383, 384, 385, 386, 387, 390, 391, 393, 395

Offset: 0

Author

Tilman Piesk, Jul 22 2013

Keywords

• nonn

Comments

Two Boolean functions belong to the same big equivalence class (bec) when they can be expressed by each other by negating and permuting arguments. E.g., when f(~p,r,q) = g(p,q,r), then f and g belong to the same bec. Geometrically this means that the functions correspond to hypercubes with binarily colored vertices that are equivalent up to rotation and reflection.

Boolean functions correspond to integers, so each bec can be denoted by the smallest integer corresponding to one of its functions. There are A000616(n) big 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 A000616(2) = 6 big equivalence classes:
a     a(n)    Boolean functions            hypercube (square)
0      0      0000                         empty
1      1      0001, 0010, 0100, 1000       one in a corner
2      3      0011, 1100, 0101, 1010       ones on a side
3      6      0110, 1001                   ones on a diagonal
4      7      0111, 1011, 1101, 1110       ones in 3 corners
5     15      1111                         full
		

Links

• Tilman Piesk, Table of n, a(n) for n = 0..9999
• Tilman Piesk, Big equivalence classes of Boolean functions
• Tilman Piesk, bec of 4-ary functions corresponding to a(85) = 854 = 0x0356
• Tilman Piesk, MATLAB code used for the calculation
• Index entries for sequences related to Boolean functions

Crossrefs

Cf. A227722 (does the same for small equivalence classes).

Cf. A000616, A051179.

Formula

a( A000616 - 1 ) = a(2,5,21,401,...) = 3,15,255,65535,... = A051179