A351900 a(n) is the smallest positive integer not the sum of one or more nonzero powers of p_0, p_1, ..., p_n, where p_0 = 1 and p_n = prime(n) for n >= 1.

2, 6, 15, 62, 112, 910, 5161, 14110, 359369, 9719153, 119512942

Offset

0,1

Comments

a(n) cannot be written as Sum_{i=0..n} c_i*p_i^k_i, where c_i = 0 or 1 and k_i is a positive integer.

Links

Table of n, a(n) for n=0..10.

Jon E. Schoenfield, Magma program

Example

a(0) = 2 because 1 = p_0 and 2 is the smallest integer not the nonzero power of p_0.
a(1) = 6 because 1 = p_0, 2 = p_1, 3 = p_0 + p_1, 4 = p_1^2, 5 = p_1^2 + p_0, and 6 is the smallest integer not the sum of nonzero powers of p_0 and p_1.

Prog

(Python)
from itertools import product, combinations; from sympy import prime
def check(M):
    for L in product(*M):
        for i in range(1, len(L)+1):
            for c in set(combinations(L, i)):
                s = sum(c); W.add(s)
                if s == m: return 1
m_max, R, N = 10**8, [2], [{1}]
for n in range(1, 10):
    N.append({prime(n)}); W = {1}
    for m in range(2, m_max):
        for i in range(len(N)):
            t = max(N[i])*min(N[i])
            while 1 < t <= m:
                if t == m: W.add(t)
                N[i].add(t); t = max(N[i])*min(N[i])
        if m in W or check(N): continue
        R.append(m); break
print(*R, sep = ', ')

Crossrefs

Cf. A034875, A122863, A351661.

Sequence in context: A003268 A242792 A216811 * A335751 A307180 A009455

Adjacent sequences: A351897 A351898 A351899 * A351901 A351902 A351903

Keyword

nonn,more

Author

Ya-Ping Lu, Feb 24 2022

Extensions

a(9)-a(10) from Jon E. Schoenfield, Feb 25 2022

Status

approved