A071848 -id:A071848 - cp's OEIS frontend

A071313 a(n) is the smallest number that cannot be obtained from the numbers {1,3,...,2n-1} using each number at most once and the operators +, -, , /, where intermediate subexpressions must be integers.

2, 5, 11, 41, 92, 733, 4337, 28972, 195098, 1797746

Offset: 1

Koksal Karakus (karakusk(AT)hotmail.com), Jun 11 2002

If noninteger subexpressions are permitted, a(5) = 122 and not 92 since 92 = (3+7)*(9 + 1/5). - Michael S. Branicky, Jul 01 2022

			a(2)=5 because using {1,3} and the four operations we can obtain 1=1, 3-1=2, 3=3, 3+1=4 but we cannot obtain 5 in the same way.

Gilles Bannay, Countdown Problem
Index entries for similar sequences

Cf. A060315, A071848.

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][(2*i-1,)] = {2*i-1}
    reach = set(range(1, 2*n, 2))
    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)
    k = 1
    while k in reach: k += 1
    return k
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 01 2022

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

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.

Original entry on oeis.org

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

Offset: 1

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

			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).

Index entries for similar sequences

Cf. A060315, A070960, A071848.

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

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

cp's OEIS Frontend

A071313 a(n) is the smallest number that cannot be obtained from the numbers {1,3,...,2n-1} using each number at most once and the operators +, -, , /, where intermediate subexpressions must be integers.

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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.

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cp's OEIS Frontend

A071313 a(n) is the smallest number that cannot be obtained from the numbers {1,3,...,2*n-1} using each number at most once and the operators +, -, *, /, where intermediate subexpressions must be integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A071313 a(n) is the smallest number that cannot be obtained from the numbers {1,3,...,2n-1} using each number at most once and the operators +, -, , /, where intermediate subexpressions must be integers.