OFFSET
1,2
COMMENTS
The shift operation here is also sometimes called successor, see A263283.
Note this complexity measure counts both operands (the ones) and operators (the shifts and multiplications), whereas most of the complexity measures in the crossrefs count only operands. However, in the presence of successor it would not make sense to count only operands, since any number can be expressed with a single occurrence of 1. - Glen Whitney, Oct 06 2021
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..67
Akshunna Shaurya Dogra, Minimal Representations of Natural Numbers Under a Set of Operators, arXiv preprint 1801.01360 [math.HO], Jan 2018.
EXAMPLE
1 = 1 has complexity 1
2 = S1 has complexity 2
3 = SS1 has complexity 3
4 = SSS1 has complexity 4
5 = SSSS1 has complexity 5
6 = SSSSS1 has complexity 6
7 = SSSSSS1 has complexity 7
10 = S*SS1SS1 = shift(product of (3 and 3)) has complexity 8
(Note that 8 = *S1SSS1 has complexity 7)
11 = SS*SS1SS1 has complexity 9
14 = SS*SS1SSS1 has complexity 10
PROG
(Python)
def aupton(nn):
alst, R, allR = [1], {1: {1}}, {1} # R[n] is set reachable using n ops
for n in range(2, nn+1):
R[n] = set(a+1 for a in R[n-1])
R[n] |= set(a*b for i in range(1, (n+1)//2) for a in R[i] for b in R[n-1-i])
alst.append(min(R[n] - allR))
allR |= R[n]
return alst
print(aupton(49)) # Michael S. Branicky, Oct 06 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 11 2018
STATUS
approved