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, 11999, 180263

Offset

1,2

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.

From Brendan J. Dee, Apr 11 2026: (Start)

This sequence is infinite: for every n>1 there are infinitely many admissible sets. The following example shows that a set can be constructed by generating numbers of sufficient bit length so that any carries are confined and cannot interfere across bit positions.

For n=10, we can construct the admissible set: {39335582261627482034321941888000, 39335582261627482034321941856287, 39335582261627482034321909382175, 39335582261627482034288655891487, 39335582261627482000237081426975, 39335582261627447131424829766687, 39335582261591741467679129631775, 39335582225029141792082191481887, 39335544784927073980817525996575, 39297206120409635245800069037087}; the migrating 10-bit zero gap is visible in hexadecimal: 0x1f07c1f07c1f07c1f07c1f07c00, 0x1f07c1f07c1f07c1f07c1f0001f, 0x1f07c1f07c1f07c1f07c0007c1f, 0x1f07c1f07c1f07c1f0001f07c1f, 0x1f07c1f07c1f07c0007c1f07c1f, 0x1f07c1f07c1f0001f07c1f07c1f, 0x1f07c1f07c0007c1f07c1f07c1f, 0x1f07c1f0001f07c1f07c1f07c1f, 0x1f07c0007c1f07c1f07c1f07c1f, 0x1f0001f07c1f07c1f07c1f07c1f.

(End)

Links

Table of n, a(n) for n=1..8.

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

Example

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.
From Brendan J. Dee, Apr 11 2026: (Start)
n=7: {768, 1440, 1953, 1954, 1956, 1960, 1968} with sum = 11999.
n=8: {12, 16471, 16982, 17494, 18518, 20566, 24662, 65558} with sum = 180263. (End)

Prog

(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]++<v[j+1] && return(ve[j]=A1969[v[j]]); ve[j]=A1969[v[j]=j])/*end for*/; ve[n]=A1969[v[n]++])/*end local()*/; while( s[n]+ve[n]<m, for(i=2, n, s[n-i+1]+sum(j=n-i+1, n, ve[j]) < m && ok(vecextract(ve, 2^n-2^(n-i))) && next; inc(i); next(2)); m>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. */
(Python)
from itertools import combinations # ok() just checks whether a set is OK
odious = lambda x: x.bit_count()&1
ok = lambda S: not any(map(odious, S)) and all(odious(sum(s)) for k in range(2, len(S)+1) for s in combinations(S, k)) # M. F. Hasler, Apr 10 2026

Crossrefs

Sequence in context: A302306 A067950 A303037 * A022273 A302874 A061294

Adjacent sequences: A230383 A230384 A230385 * A230387 A230388 A230389

Keyword

nonn,more,hard

Author

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

Extensions

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

a(7) from Donovan Johnson, Oct 27 2013

a(7) corrected and a(8) added by Brendan J. Dee, Apr 07 2026

Status

approved