A230386 Least sum of a set of n evil numbers (A001969) such that any two or more add to an odious number (A000069).

0, 8, 31, 64, 191, 1797, 18463

Offset: 1

Author

Vladimir Shevelev and M. F. Hasler, Oct 17 2013

Keywords

• nonn
• more
• hard

Comments

Row sums of A230385: see there for the corresponding sets. See A230387 for a "dual" version.

Is this sequence finite, or is there for any n at least one admissible set of n evil numbers, i.e., such that any sum of two or more elements add up to an odious number?

By definition, this is a subsequence of the odious numbers A000069.

Examples

			The table A230385 reads
n=1: {0} with sum = 0,
n=2: {3, 5} with sum = 8,
n=3: {5, 9, 17} or {9, 10, 12} with sum = 31,
n=4: {5, 9, 17, 33} with sum = 64,
n=5: {33, 34, 36, 40, 48} with sum = 191,
n=6: {257, 264, 278, 288, 326, 384} with sum = 1797.
For example, for n=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.
n=7: {801, 802, 804, 808, 816, 4896, 9536} with sum = 18463.
		

Links

• M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013

Programs

PARI

(is_A69=n->bittest(hammingweight(n),0)); A1969=select(n->!is_A69(n), vector(1600,n,n)) /* no 0 here! */; A230386(n,m=9e9)={ local(v=vector(n,i,i), ve=vector(n,i,A1969[i]), t=0, s=vector(n,i,if(i>1,A230386(i-1))), S(v)=sum(j=1,#v,v[j]), ok(e)=!forstep(i=3,2^#e-1,2, is_A69( S( vecextract( e,i )))||return), inc(i)=for(j=1,n-i,v[j]=j); for(j=n-i+1,n-1, v[j]++S(ve) && /*print*/([m=S(ve),ve]); inc(n));m} /* This code is very fast up to n=5 and much too slow for n>5. */

Extensions

a(6) added by M. F. Hasler, Oct 18 2013

a(7) from Donovan Johnson, Oct 27 2013