

A227723


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


5



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
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OFFSET

0,3


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 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



FORMULA



EXAMPLE

The 16 2ary 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


CROSSREFS

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


KEYWORD

nonn


AUTHOR



STATUS

approved



