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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
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
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
KEYWORD
nonn,more
AUTHOR
Ya-Ping Lu, Feb 24 2022
EXTENSIONS
a(9)-a(10) from Jon E. Schoenfield, Feb 25 2022
STATUS
approved