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

Fri Oct 18 16:27:07 CEST 2013

```Number N(n) of conditions on elements of required sets too fast grows:
N(n) = 2^n - n - 1.  In particular, N(5) = 26, while N(6) = 57. Maybe, the system  of  57 restrictions is a "critical mass", but I think that it is unlikely and continue to believe that this sequence (as also other 3 conjugate ones) is infinite.

Best regards,

----- Original Message -----
From: "M. F. Hasler" <oeis at hasler.fr>
Date: Thursday, October 17, 2013 10:10
Subject: [seqfan] Re: Peculiar sets of evil numbers (Cf. A001969)
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

>
> 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, , ]
> [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,
> >
> >
> > _______________________________________________
> >
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