[seqfan] Re: Peculiar sets of evil numbers (Cf. A001969)

M. F. Hasler oeis at hasler.fr
Fri Oct 18 00:10:12 CEST 2013

```Dear Vladimir & SeqFans,

I wrote a little script which confirms your solutions for n=2,3,4 and
finds for n=5 the solution
[a(5) = 191, E_n = [33, 34, 36, 40, 48], indices = [16, 17, 18, 20, 24]]
in less than 0.2 seconds,
but it does not find a solution for n=6.
(Maybe I was not patient enough, I killed it after about a minute
without any output,
while for n=4 it displays some non-optimal solutions before finding
the best one.)

For the opposite case (peculiar sets of odious numbers, such that any
partial sum is evil), I get:

[ a(n)=sum, O_n, indices ]
[1, [1], [1]]
[3, [1, 2], [1, 2]]
[17, [2, 7, 8], [2, 4, 5]]
[139, [4, 19, 49, 67], [3, 10, 25, 34]]

and again that last one is found in ~ 0.1 sec, but the next one is not
found at all.

Best regards,
Maximilian

On Tue, Oct 15, 2013 at 7:18 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
>
> Dear Seqfans,
>
> Let E_n={e_1,e_2,...,e_n} be a set of distinct evil numbers e_i (Cf. A001969), such that sum of all elements of every subset containing >=2 elements, is odious (Cf. A000069). Let us name E_n a peculiar set of evils.
> Let sequence: a(n) be the smallest possible sum of elements of a peculiar set of n evil  numbers (n>=2).
> By handy, I find  a(2)=8, which corresponds to E_2={3,5},  a(3)=31, which corresponds  to E_3={5,9,17}, a(4)=64, which corresponds to E_4={5,9,17,33}. For example, for E_4, all 11 numbers 5+9=14,5+17=22,5+33=38,9+17=26,9+33=42,17+33=50, 5+9+17=31,5+9+33=47,5+17+33=55,9+17+33=59, 5+9+17+33=64 are odious.
> Can anyone continue this sequence (one can conjecture that it is infinite)?
> A dual problem one can pose for O_n={o_1,...,o_n} such that o_i are odious (A000069), while sums of elements of every subset  with >=2 elements  are evil. For example, for set {7,11,22} of odious numbers, we have 7+11=18, 7+22=29, 11+22=33, 7+11+22=40 all are evil.
>
> Best regards,