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.
3, 5, 6, 8, 10, 12, 13, 17, 19, 20, 22, 27, 32, 34, 36, 37, 41, 43, 44, 46, 61, 67, 68, 82, 84, 91, 95, 107, 119, 126, 129, 131, 153, 167, 204, 211, 214, 252, 261, 416, 452, 489, 499, 537, 6006, 6265, 6266, 6312, 190852, 207403, 208524, 208806, 211967, 213074, 213594, 213677, 214781, 215042, 215075, 215077
Offset: 1
Keywords
Examples
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).
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.
Crossrefs
Cf. A297345.
Programs
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Python
# generates all elements of the sequence, smaller than 6268 import numpy as np import itertools def g(i,s,perms): c = 0 for iks in perms: t=np.asarray(iks) if np.dot(t,s) == i: c += 1 if c == 2: break if c == 1: print(i) S=[1, 2, 7,24,85,285,1143] S1=[0,1, 2, 7,24,85,285,1143] perms = [p for p in itertools.product(S1, repeat=len(S))] s=np.asarray(S,dtype=np.int64) for i in range(1,6268): if i not in S: g(i,s,perms)