A071603 Number of different positive integers that we can obtain from the integers {1,2,...,n} using each number at most once and the operators +, -, *, /, where intermediate subexpressions must be integers.

1, 3, 9, 31, 121, 542, 2868, 16329, 106762, 758155, 6142570

Offset: 1

Author

Koksal Karakus (karakusk(AT)hotmail.com), Jun 02 2002

Keywords

• more
• nonn

Examples

			a(4)=31 because we can obtain the positive integers 1,2,...,28 and 30,32,36 by using the integers {1, 2, 3, 4} at most once and the four operations. For example 30 = 3*2*(4+1).
		

Links

• Index entries for similar sequences

Crossrefs

Cf. A060315, A070960, A071848.

Programs

Python

def a(n):
    R = dict() # index of each reachable subset is [card(s)-1][s]
    for i in range(n): R[i] = dict()
    for i in range(1, n+1): R[0][(i,)] = {i}
    reach = set(range(1, n+1))
    for j in range(1, n):
        for i in range((j+1)//2):
            for s1 in R[i]:
                for s2 in R[j-1-i]:
                    if set(s1) & set(s2) == set():
                        s12 = tuple(sorted(set(s1) | set(s2)))
                        if s12 not in R[len(s12)-1]:
                            R[len(s12)-1][s12] = set()
                        for a in R[i][s1]:
                            for b in R[j-1-i][s2]:
                                allowed = [a+b, a*b, a-b, b-a]
                                if a!=0 and b%a==0: allowed.append(b//a)
                                if b!=0 and a%b==0: allowed.append(a//b)
                                R[len(s12)-1][s12].update(allowed)
                                reach.update(allowed)
    return len(set(r for r in reach if r > 0 and r.denominator == 1))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 01 2022

Extensions

a(10)-a(11) from Michael S. Branicky, Jul 01 2022