arrays - find a list of integers for a checksum -


i need list of n positive integers l has properties:

  • for each possible subset s of l, if sum items of s, sum not in l
  • for each possible subset s of l, if sum items of s, sum unique (each subset can identified sum)

working example 1:

n = 4 l = [1, 5, 7, 9]  check: 1+5   = 6  ok 5+7   = 12 ok 7+9   = 16 ok 9+1   = 10 ok 1+7   = 8  ok 5+9   = 14 ok  1+5+7 = 13 ok 5+7+9 = 21 ok 1+5+9 = 15 ok 1+7+9 = 17 ok  1+5+7+9 = 22 ok  sums unique -> l ok n = 4

as easy construct sequence, suggest using power series, e.g.

 1, 2,  4,  8, ..., 2**k, ...  1, 3,  9, 27, ..., 3**k, ...   1, 4, 16, 64, ..., 4**k, ...  ...  1, n, n**2, n**3,..., n**k, ... n >= 2

take, instance, 2: neither power of 2 sum of other 2 powers; given sum (number) can find out subset converting sum binary representation:

 23 = 10111 (binary) = 2**0 + 2**1 + 2**2 + 2**4 = 1 + 2 + 4 + 16

in general case, simple greedy algorithm do: given sum subtract largest item less or equal sum; continue subtracting zero:

 n = 3  sum = 273   273 - 243 (3**5) = 30   30 -  27 (3**3) =  3    3 -   3 (3**1) =  0  273 = 3**5 + 3**3 + 3**1 = 243 + 27 + 3

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