A003064 -id:A003064 - cp's OEIS frontend

A075530 a(n) is the largest m such that n+1, n+2, ..., n+m are all members of the set X(n) given in A075529.

1, 1, 2, 3, 6, 13, 22, 39, 62, 117, 180, 367, 594, 1073, 1888, 3567

Offset: 0

Eli Ben-Naim (ebn(AT)lanl.gov), Sep 19 2002

			X(1) = {2} hence a(1)=1, X(2) = {3,4} hence a(2)=2,  X(3) = {4,5,6,8} hence a(3)=3.

E. Ben-Naim and P. L. Krapivsky, Growth and Structure of Stochastic Sequences, J. Phys. A (2002).
E. Ben-Naim, Home page

Numbers so far suggest that a(n) = A003064(n) - n + 1, n>1. - Ralf Stephan, Mar 21 2004

Entry revised by Sean A. Irvine, Feb 20 2025

A137814 Smallest size of a topology that needs at least n points.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 29, 47, 79, 127, 191, 379

Offset: 0

Bridget Tenner, Feb 11 2008

nonn

			There is no topology with less than 4 points having 7 open sets. However, there do exist topologies on 3 points that have 2, 3, 4, 5, 6 and 8 open sets.

M. Erné and K. Stege, Counting finite posets and topologies, Tech. Report 236, University of Hannover, 1990.

Swee Hong Chan and Igor Pak, Computational complexity of counting coincidences, arXiv:2308.10214 [math.CO], 2023. See p. 10.
M. Erné and K. Stege, Counting finite posets and topologies, Order, September 1991, Volume 8, Issue 3, pp 247-265.
K. Ragnarsson and B. E. Tenner, Obtainable sizes of topologies on finite sets, J. Combin. Theory Ser. A 117 (2010) 138-151.

Cf. A137813 and A003064 (smallest number which needs an addition chain of at-least-length n).

Name improved and a(0), a(1), a(12) added by Achim Flammenkamp, Oct 23 2016

A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 5, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 5, 4, 6, 5, 4, 6, 4, 5, 5, 5, 5, 5, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 6, 4, 6, 7, 5, 6, 7, 5, 6, 6, 5, 5, 7, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5

Offset: 1

Tatsuru Murai, Aug 08 2003

nonn

n = 12509 is the first n for which a(n) = 0 because it is the smallest number that has no shortest addition chain of Brauer type. - Hugo Pfoertner, Jun 10 2006 [Edited by Pontus von Brömssen, Apr 25 2025]

			a(23)=5 because 23=1+1+2+1+4+9+5 is the shortest addition chain for 23.
For n=9 there are A079301(9)=3 different shortest addition chains, all of Brauer type:
[1 2 3 6 9] -> 9=1+1+1+3+3 -> 2 different addends {1,3}
[1 2 4 5 9] -> 9=1+1+2+1+4 -> 3 different addends {1,2,4}
[1 2 4 8 9] -> 9=1+1+2+4+1 -> 3 different addends {1,2,4}
The minimum number of different addends is 2, therefore a(9)=2.

Giovanni Resta, Tables of Shortest Addition Chains, computed by David W. Wilson.
Index to sequences related to the complexity of n

Cf. A003064, A003065, A003313, A005766, A008057, A008928, A008933, A079300, A079300, A079301, A349044.

a(n) = 0 if and only if n is in A349044. - Pontus von Brömssen, Apr 25 2025

Edited by Hugo Pfoertner, Jun 10 2006

Escape clause added by Pontus von Brömssen, Apr 25 2025

cp's OEIS Frontend

A075530 a(n) is the largest m such that n+1, n+2, ..., n+m are all members of the set X(n) given in A075529.

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A137814 Smallest size of a topology that needs at least n points.

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A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

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