A007600 Minimal number of subsets in a separating family for a set of n elements.

0, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, and Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

Let j = ceiling(log_3(n))-1. Then a(n) = 3j+1 if 3^j < n <= 4*3^(j-1); 3j+2 if 4*3^(j-1) < n <= 2*3^j; 3j+3 if 2*3^j < n <= 3^(j+1). - Ralf Stephan, Apr 28 2003

"In 1973, The Hungarian mathematician G. O. H. Katona posed the general problem of determining, for a set of n elements, the minimum number f(n) of subsets in a separating family. This problem was solved in early Feb, 1982, by the gifted Chinese mathematician Cai Mao-Cheng (Academia Sinica, Peking), during an extended visit to the University of Waterloo." [Honsberger]

Honsberger gives a misattribution: the problem was first solved by Andrew Chi-Chih Yao. - Vincent Vatter, Apr 24 2006

References

• Ross Honsberger, Mathematical Gems III, Dolciani Mathematical Expositions No. 9, Mathematical Association of America, 1985, Cai Mao-Cheng's Solution to Katona's Problem on Families of Separating Subsets, Chapter 18, pages 224 - 239.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, preprint, 2003.
• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
• M-C. Cai, Solutions to Edmonds' and Katona's problems on families of separating sets, Discrete Math., 47 (1983) 13-21.
• Gyula O. H. Katona, Home page.
• Jeffrey Shallit, k-regular Sequences, Colloque Mendès-France, Bordeaux 12 September 2000.
• Vincent Vatter, Maximal independent sets and separating covers, Amer. Math. Monthly, 118 (2011), 418-423. [_N. J. A. Sloane_, May 05 2011]
• Eric Weisstein's World of Mathematics, Katona's Problem.
• Eric Weisstein's World of Mathematics, Separating Family.
• A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.

Crossrefs

Positions of increases are in A007601. This is a left inverse of A000792.

Programs

Maple

for n from 1 to 5 do C[n]:=n; od; C[6]:=5;
for i from 2 to 5 do
   for m from 2*3^(i-1)+1 to 3^i do C[m]:=3*i; od:
   for m from 3^i+1 to 4*3^(i-1) do C[m]:=3*i+1; od:
   for m from 4*3^(i-1)+1 to 2*3^i do C[m]:=3*i+2; od:
od:
[seq(C[i],i=1..120)]; # N. J. A. Sloane, May 05 2011

Mathematica

f[n_] := Min[ Table[2p + 3Ceiling[Log[3, n/2^p]], {p, 0, 2}]]; Table[ f[n], {n, 80}] (* Robert G. Wilson v, Jan 15 2005 *)

PARI

a(n) = vecmin(vector(3, i, my(k=i-1); 2*k + 3*ceil(log(n/2^k)/log(3)))); \\ Michel Marcus, Dec 18 2022

Formula

a(n) = Min_{k=0..2} 2*k + 3*ceiling(log_3(n/2^k)).

a(A000792(n)) = n, for n>1; Andrew Chi-Chih Yao attributes this observation to D. E. Muller. - Vincent Vatter, Apr 24 2006

a(n) = ceiling(3*log_3(n)) if n belongs to A000792. - Mehmet Sarraç, Dec 17 2022