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

1,1

Links

Table of n, a(n) for n=1..60.

Example

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.

Prog

(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.int)
for i in range(1, 6268):
   if i not in S:
       g(i, s, perms)

Crossrefs

Cf. A297345.

Sequence in context: A329841 A047329 A187685 * A191777 A284590 A287724

Adjacent sequences: A300727 A300728 A300729 * A300731 A300732 A300733

Keyword

nonn

Author

Luis F.B.A. Alexandre, Mar 11 2018

Status

approved