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

1,2

Links

Table of n, a(n) for n=1..34.

Index entries for similar sequences

Example

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.

Prog

(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

Crossrefs

Cf. A048249.

Sequence in context: A096236 A356932 A002574 * A090427 A006745 A049284

Adjacent sequences: A069762 A069763 A069764 * A069766 A069767 A069768

Keyword

nonn,more

Author

John W. Layman, Apr 05 2002

Extensions

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

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

Status

approved