A349044 -id:A349044 - cp's OEIS frontend

A230697 Length of shortest addition-multiplication chain for n.

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 4, 5, 4, 5, 5, 5, 6, 4, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 5, 6, 6, 5, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 5, 6, 6, 6, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 5, 6, 6, 6, 7, 5, 4, 5, 5, 5, 6, 6, 6, 6

Offset: 1

Harry Altman, Oct 27 2013

			A shortest addition-multiplication chain for 16 is (1,2,4,16), of length a(16) = 3.
A shortest addition-multiplication chain for 281 is (1,2,4,5,16,25,256,281), of length a(281) = 7. This is the first case where not all terms in some shortest chain are the sum or product of the immediately preceding term and one more preceding term. In other words, 281 is the smallest of the analog of non-Brauer numbers (A349044) for addition-multiplication chains. The next ones are 913, 941, 996, 997, 998, 1012, 1077, 1079, 1542, 1572, 1575, 1589, 1706, 1792, 1795, 1816, 1864, ... . - _Pontus von Brömssen_, May 02 2025

Jinyuan Wang, Table of n, a(n) for n = 1..10000
H. M. Bahig, On a generalization of addition chains: Addition-multiplication chains, Discrete Mathematics 308 (2008), 611-616.
Gleb Ivanov, Python program.

Cf. A003313, A005245, A128998, A173419, A349044, A354914, A383001, A383002.

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

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A230697 Length of shortest addition-multiplication chain for n.

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