A069765 Number of distinct values obtained using n ones and the operations of sum, product and quotient.

1, 2, 4, 7, 13, 24, 42, 77, 138, 249, 454, 823, 1493, 2719, 4969, 9060, 16588, 30375, 55672, 102330, 188334, 346624, 639280, 1179742, 2178907, 4029060, 7456271, 13806301, 25587417, 47452133, 88057540, 163518793, 303826088, 564825654

Offset: 1

Author

John W. Layman, Apr 05 2002

Keywords

• nonn
• more

Examples

			a(5)=13 because five ones yield the following 13 distinct values and no others: 1+1+1+1+1=5, 1+1+1+(1/1)=4, 1/(1+1+1+1)=1/4, 1+(1/1)+(1/1)=3, 1/(1+1+(1/1))=1/3, 1+(1/(1+1+1))=4/3, 1+(1/1)*(1/1)=2, 1/((1/1)+(1/1))=1/2, (1+1+1)/(1+1)=3/2, 1+1+(1/(1+1))=5/2, (1+1)/(1+1+1)=2/3, 1*1*1*1*1=1 and (1+1)*(1+1+1)=6.
		

Links

• Index entries for similar sequences

Crossrefs

Cf. A048249.

Programs

Python

from fractions import Fraction
from functools import lru_cache
@lru_cache()
def f(m):
    if m == 1: return {Fraction(1, 1)}
    out = set()
    for j in range(1, m//2+1):
        for x in f(j):
            for y in f(m-j):
                out.update([x + y, x * y])
                if y: out.add(Fraction(x, y))
                if x: out.add(Fraction(y, x))
    return out
def a(n): return len(f(n))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jul 28 2022

Extensions

a(20)-a(30) from Michael S. Branicky, Jul 29 2022

a(31)-a(34) from Michael S. Branicky, Jun 30 2023