A351900 - OEIS

%I #13 Feb 26 2022 08:38:00

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

%N 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.

%C 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.

%H Jon E. Schoenfield, <a href="/A351900/a351900_1.txt">Magma program</a>

%e a(0) = 2 because 1 = p_0 and 2 is the smallest integer not the nonzero power of p_0.

%e 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.

%o (Python)

%o from itertools import product, combinations; from sympy import prime

%o def check(M):

%o     for L in product(*M):

%o         for i in range(1, len(L)+1):

%o             for c in set(combinations(L, i)):

%o                 s = sum(c); W.add(s)

%o                 if s == m: return 1

%o m_max, R, N = 10**8, [2], [{1}]

%o for n in range(1, 10):

%o     N.append({prime(n)}); W = {1}

%o     for m in range(2, m_max):

%o         for i in range(len(N)):

%o             t = max(N[i])*min(N[i])

%o             while 1 < t <= m:

%o                 if t == m: W.add(t)

%o                 N[i].add(t); t = max(N[i])*min(N[i])

%o         if m in W or check(N): continue

%o         R.append(m); break

%o print(*R, sep = ', ')

%Y Cf. A034875, A122863, A351661.

%K nonn,more

%O 0,1

%A _Ya-Ping Lu_, Feb 24 2022

%E a(9)-a(10) from _Jon E. Schoenfield_, Feb 25 2022