This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A305233 #64 Aug 11 2025 04:42:18
%S A305233 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,
%T A305233 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,
%U A305233 8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A305233 Smallest k such that binomial(k, floor(k/2)) >= n.
%C A305233 Minimal size of a set with n subsets such that no one contains another.
%C A305233 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
%C A305233 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
%D A305233 K. H. Kim, Boolean matrix theory and applications. Marcel Dekker, New York and Basel (1982).
%H A305233 David A. Corneth, <a href="/A305233/b305233.txt">Table of n, a(n) for n = 1..10000</a>
%H A305233 Dmitry I. Ignatov and Yazag Meziane <a href="/A305233/a305233.pdf">An asymptototic lower bound for A305233</a>.
%H A305233 Evgeny E. Marenich, <a href="https://doi.org/10.1142/9789812836021_0005">Determining the Schein Rank of Boolean Matrices</a>. Matrix Methods: Theory, Algorithms and Applications (2010) 85-103.
%H A305233 Emanuel Sperner, <a href="https://gdz.sub.uni-goettingen.de/id/PPN266833020_0027">Ein Satz über Untermengen einer endlichen Menge</a>, Mathematische Zeitschrift 27, 544-548, (1928).
%F A305233 Asymptotic lower bound is log_2(sqrt(Pi/2)*n) + O(log_2(log_2(sqrt(Pi/2)*n))). - _Dmitry I. Ignatov_ and _Meziane Yazag_, Dec 07 2023
%t A305233 Array[Block[{k = 1}, While[Binomial[k, Floor[k/2]] < #, k++]; k] &, 105] (* _Michael De Vlieger_, Aug 02 2018 *)
%o A305233 (Python)
%o A305233 from sympy import binomial
%o A305233 def f(n): return binomial(n, n // 2)
%o A305233 sum([[i]*(f(i)-f(i-1)) for i in range(1,10)],[1])
%o A305233 (PARI) a(n) = my(k=1); while (binomial(k, floor(k/2)) < n, k++); k; \\ _Michel Marcus_, Jul 10 2018
%o A305233 (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[i-1] + 1, min(n,l[i]), res[j]=i)); res \\ _David A. Corneth_, May 22 2020
%Y A305233 Cf. A001405, A056641.
%K A305233 nonn
%O A305233 1,2
%A A305233 _Jack Zhang_, May 28 2018