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A230387
Least sum of a set of n odious numbers (A000069) such that the sum of two or more is an evil number (A001969).
4
1, 3, 17, 139, 795, 3903, 28575
OFFSET
1,2
COMMENTS
Is this sequence finite, or is there for any n at least one admissible set of n odious numbers, i.e., such that any sum of two or more elements add up to an evil number?
LINKS
M. F. Hasler, in reply to V. Shevelev, Peculiar sets of evil numbers (Cf. A001969), SeqFan list, Oct 17 2013
FORMULA
Row sums of A230384.
EXAMPLE
For n=1 to 4, we have the sets
n=1: {1} with sum = 1,
n=2: {1, 2} with sum = 3
n=3: {2, 7, 8} with sum = 17,
n=4: {4, 19, 49, 67} with sum = 139.
E.g., for n=3, the numbers 2, 7 and 8 have an odd bit sum, but 2+7, 2+8, 7+8 and 2+7+8 all have an odd bit sum.
For n=4, we also have the admissible set {14, 31, 44, 61} which has a smaller maximal element, but a larger total sum.
n=5: {42, 84, 138, 174, 357} with sum = 795.
n=6: {168, 348, 372, 702, 906, 1407} with sum = 3903.
n=7: {2273, 2274, 2276, 2280, 2288, 3296, 13888} with sum = 28575.
PROG
(PARI) A69=select(is_A69=n->bittest(hammingweight(n), 0), vector(700, n, n)); A230387(n, m=9e9)={ local(v=vector(n, i, i), ve=vector(n, i, A69[i]), t=0, s=vector(n, i, if(i>1, A230387(i-1))), ok(e)=!forstep(i=3, 2^#e-1, 2, is_A69( sum( j=1, #t=vecextract(e, i), t[j] )) && 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]=A69[v[j]]); ve[j]=A69[v[j]=j])/*end for*/; ve[n]=A69[v[n]++]); /*end of 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=min(sum(j=1, n, ve[j]), m); inc(n)); m} /* Note: The code may find a(n) earlier but won't return it unless A69 is precomputed up to the limit a(n) - a(n-1); so e.g. 700 is enough for a(5).*/
CROSSREFS
Sequence in context: A286896 A244432 A219503 * A370767 A361626 A360583
KEYWORD
nonn,base,more,hard
AUTHOR
M. F. Hasler, Oct 17 2013
EXTENSIONS
a(5)-a(6) from Charles R Greathouse IV, Oct 18 2013
a(7) from Donovan Johnson, Oct 27 2013
STATUS
approved