

A305233


Smallest k such that binomial(k, floor(k/2)) >= n.


1



1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
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OFFSET

1,2


COMMENTS

Minimal size of a set with n subsets such that no one contains another.
The proof that the minimal cardinality k of a set having n subsets not containg one other is the generalized central binomial coefficient binomial(k, floor(k/2)(equivalent to: "the largest possible families of finite sets none of which contain any other sets in the family) is called "Sperner's Theorem" and is due to Sperner  Renzo Benedetti, May 26 2021
Also the Schein rank of a contranominal scale of size n represented as a Boolean matrix (Kim, 1982; Marenich, 2010).  Dmitry I. Ignatov, Nov 25 2021


REFERENCES

K. H. Kim, Boolean matrix theory and applications. Marcel Dekker, New York and Basel (1982).


LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000
Evgeny E. Marenich, Determining the Schein Rank of Boolean Matrices. Matrix Methods: Theory, Algorithms and Applications (2010) 85103.
Emanuel Sperner, Ein Satz über Untermengen einer endlichen Menge, Mathematische Zeitschrift 27, 544548, (1928).


MATHEMATICA

Array[Block[{k = 1}, While[Binomial[k, Floor[k/2]] < #, k++]; k] &, 105] (* Michael De Vlieger, Aug 02 2018 *)


PROG

(Python)
from sympy import binomial
def f(n): return binomial(n, n // 2)
sum([[i]*(f(i)f(i1)) for i in range(1, 10)], [1])
(PARI) a(n) = my(k=1); while (binomial(k, floor(k/2)) < n, k++); k; \\ Michel Marcus, Jul 10 2018
(PARI) first(n) = my(l=List(), t=1, res = vector(n), c=1); while(c<=n, listput(l, c); t++; c=binomial(t, t\2)); listput(l, c); res[1]=1; for(i=2, #l, m = max(n, l[i]); for(j=l[i1] + 1, min(n, l[i]), res[j]=i)); res \\ David A. Corneth, May 22 2020


CROSSREFS

Cf. A001405.
Sequence in context: A276334 A225486 A325282 * A130242 A130245 A087793
Adjacent sequences: A305230 A305231 A305232 * A305234 A305235 A305236


KEYWORD

nonn,changed


AUTHOR

Jack Zhang, May 28 2018


STATUS

approved



