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

M. F. Hasler oeis at hasler.fr
Fri Oct 18 22:21:34 CEST 2013

```Charles Greathouse found two more terms for the dual sequence I
proposed as https://oeis.org/draft/A230387, and I suppose he could
also find at least 1 or 2 more terms for this one.
Best regards,
Maximilian

On Fri, Oct 18, 2013 at 10:27 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> 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, [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,
>> >
>> >
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>> >
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>