A181957 Smallest positive integer which cannot be calculated by an expression containing n binary operators (either add or multiply) whose operands are integers between 1 and 9; parenthesis allowed.

10, 19, 92, 239, 829, 2831, 10061, 38231, 189311, 621791, 2853533, 11423579

Offset: 0

Author

Derek M. Jones, Apr 03 2012

Keywords

• nonn
• more

Examples

			a(4) = 239 because at least 4 operators are needed to calculate this value, e.g., (5*5+9)*7+1.
		

Links

• Derek M. Jones, R program for calculating values
• Index to sequences related to the complexity of n

Crossrefs

Cf. A181898, A181958, A181959, A181960.

Cf. A005520 (operand literal is always 1).

Programs

PARI

first(n)=my(op=[(x, y)->x+y, (x, y)->x*y], v=vector(n+1), t); v[1]=[1..9]; for(k=2, #v, my(u=[]); for(i=1, (k+1)\2, my(a=v[i], b=v[k-i]); t=Set(concat(apply(f->setbinop(f, a, b), op))); u=concat(u, t)); v[k]=setminus(Set(u), [0])); t=10; for(i=1, #v, while(setsearch(v[i], t), t++); v[i]=t); v;
print(first(7)) \\ Michael S. Branicky, Oct 19 2021 after Charles R Greathouse IV in A181898

Python

def aupton(nn):
    alst = [10]
    R = {0: set(range(1, 10))}   # R[n] is set reachable using n ops
    for n in range(1, nn):
        R[n] = set()
        for i in range((n+1)//2):
            for a in R[i]:
                for b in R[n-1-i]:
                    R[n].update([a+b, a*b])
        k = 10
        while k in R[n]: k += 1  # n.b. R[n-1] <= R[n] due to * by 1
        alst.append(k)
    return alst
print(aupton(9)) # Michael S. Branicky, Oct 19 2021

Extensions

a(5)-a(7) corrected by and a(8)-a(11) from Michael S. Branicky, Oct 19 2021