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A300730 Positive integers j of the form Sum_{i=1..k} b(i)c(i), i.e., not in A297345 such that there is only one set {c(1),...,c(k)} where the c(i) are drawn with repetition from {b(0),...,b(k)} and b(k+1) is the smallest element of A297345 that is larger than j, where b() is A297345.

%I A300730 #39 Jun 10 2025 12:26:57
%S A300730 3,5,6,8,10,12,13,17,19,20,22,27,32,34,36,37,41,43,44,46,61,67,68,82,
%T A300730 84,91,95,107,119,126,129,131,153,167,204,211,214,252,261,416,452,489,
%U A300730 499,537,6006,6265,6266,6312,190852,207403,208524,208806,211967,213074,213594,213677,214781,215042,215075,215077
%N A300730 Positive integers j of the form Sum_{i=1..k} b(i)c(i), i.e., not in A297345 such that there is only one set {c(1),...,c(k)} where the c(i) are drawn with repetition from {b(0),...,b(k)} and b(k+1) is the smallest element of A297345 that is larger than j, where b() is A297345.
%e A300730 The first positive integer not in b() is 3. To check if 3 is a(1) we note that the smallest element of b() larger than 3 is b(3)=7, hence k=2. There is only one set of coefficients {c(1),c(2)} that allows 3 to be obtained from Sum_{i=1..k} b(i)c(i). These are c(1)=2 and c(2)=1. So 3 is in fact a(1).
%e A300730 The next integer not in b() is 4. To see if it is a(2) we note that k is still 2 in this case. Now there are two possible sets of coefficients that allow the representation of 4: {0,2} and {2,1}, so 4 is not a term.
%o A300730 (Python)
%o A300730 # generates all elements of the sequence, smaller than 6268
%o A300730 import numpy as np
%o A300730 import itertools
%o A300730 def g(i,s,perms):
%o A300730    c = 0
%o A300730    for iks in perms:
%o A300730        t=np.asarray(iks)
%o A300730        if np.dot(t,s) == i:
%o A300730            c += 1
%o A300730        if c == 2:
%o A300730            break
%o A300730    if c == 1:
%o A300730        print(i)
%o A300730 S=[1, 2, 7,24,85,285,1143]
%o A300730 S1=[0,1, 2, 7,24,85,285,1143]
%o A300730 perms = [p for p in itertools.product(S1, repeat=len(S))]
%o A300730 s=np.asarray(S,dtype=np.int64)
%o A300730 for i in range(1,6268):
%o A300730    if i not in S:
%o A300730        g(i,s,perms)
%Y A300730 Cf. A297345.
%K A300730 nonn
%O A300730 1,1
%A A300730 _Luis F.B.A. Alexandre_, Mar 11 2018