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

Vladimir Shevelev shevelev at bgu.ac.il
Tue Oct 15 13:18:48 CEST 2013

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,

 Shevelev Vladimir‎

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