|
| |
|
|
A050294
|
|
Maximum cardinality of a 3-fold-free subset of {1, 2, ..., n}.
|
|
2
|
|
|
|
1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 48, 49, 49, 50, 51, 51, 52, 53, 54, 55
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
For a given r>1, a set is r-fold-free if it does not contain any subset of the form {x, r*x}.
If r is in A050376, then an r-fold-free set with the highest cardinality is obtained by removing from {1,...,n} all numbers for which r is an infinitary divisor (for the definition of the infinitary divisor of n, see comment to A037445). In general, an r-fold-free set with the highest cardinality is obtained by removing from {1,...,n} all numbers for which r is an oex divisor (for the definition of the oex divisor of n, see A186643). - Vladimir Shevelev Feb 22 2011 and Feb 28 2011.
|
|
|
LINKS
|
Bruce Reznick, Problem 1440, Mathematics Magazine, Vol. 67 (1994).
|
|
|
FORMULA
|
Take r = 3 in a(n) = (r n + sum [k = 0 to m] (-1)^k b(k)) / (r + 1), where [b(m) b(m-1) ... b(0)] is the base-r representation of n. - Rob Pratt, Apr 21 2004
Take r=3 in a(n) = n-a(floor(n/r)); a(n)=n-floor(n/r)+floor(n/r^2)-floor(n/r^3)+... [Vladimir Shevelev, Feb 22 2011].
|
|
|
EXAMPLE
|
a(26)=26-a(floor(26/3))=26-a(8)=26-6=20.
|
|
|
PROG
|
(PARI) a(n)=if(n==0, 0, n-a(floor(n/3))); \\ Joerg Arndt, Apr 27 2013
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
