|
| |
|
|
A208746
|
|
Size of largest subset of [1..n] containing no three terms in geometric progression.
|
|
8
|
|
|
|
1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 48, 49, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57, 57, 57, 58, 59, 60, 61, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 71, 72, 73, 74, 74, 75, 75, 75, 75
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
All three-term geometric progressions must be avoided, even those such as 4,6,9 whose ratio is not an integer.
David Applegate's computation used a floating-point IP solver for the packing subproblems, so although it's almost certainly correct there is no proof. First he enumerated geometric progressions using
for (i=1;i<=N;i++) {
for (j=2; j*j<=i; j++) {
if (i % (j*j) != 0) continue;
for (k=1; k<j; k++) {
print i*k*k/(j*j), i*k/j, i;
}
}
}
and then solved the integer program of maximizing the subset of {1..N} subject to not taking all 3 of any progression.
|
|
|
LINKS
|
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..125
K. O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7.
Index entries for non-averaging sequences
|
|
|
MAPLE
|
# Maple program for computing the n-th term from Robert Israel:
A:= proc(n)
local cons, x;
cons:=map(op, {seq(map(t -> x[t]+x[b]+x[b^2/t]<=2,
select(t -> (t<b) and (t>=b^2/n),
numtheory:-divisors(b^2))), b=2..n-1)});
Optimization:-Maximize(add(x[i], i=1..n), cons, assume=binary)[1]
end proc;
|
|
|
CROSSREFS
|
Cf. A003002.
Sequence in context: A053207 A334714 A138467 * A230490 A247983 A127036
Adjacent sequences: A208743 A208744 A208745 * A208747 A208748 A208749
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
David Applegate and N. J. A. Sloane, Mar 01 2012
|
|
|
EXTENSIONS
|
a(1)-a(82) confirmed by Robert Israel and extended to a(100), Mar 01 2012
|
|
|
STATUS
|
approved
|
| |
|
|
