OFFSET
1,2
COMMENTS
The poset of the positive integers is defined by covering relations "n covers (n - n/p)" for all divisors p of n.
n appears A332809(n) times in row n.
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Wikipedia, Semilattice
FORMULA
T(n,1) = T(n,n) = n. T(n, 2) = n for n >= 2.
T(x,y) <= lcm(x,y) for any x,y because x is in same chain with lcm(x,y), and y is in same chain with lcm(x,y).
Moreover, empirically it looks like T(x,y) divides lcm(x,y).
EXAMPLE
The interval [1,15] illustrates that, for example, T(12, 10) = T(6, 5) = 15, T(12, 4) = 12, T(8, 5) = 10, T(3, 1) = 3, etc.
15
_/ \_
/ \
10 12
| \_ _/ |
| \ / |
5 8 6
\_ | _/|
\_|_/ |
4 3
| _/
|_/
2
|
|
1
Triangle begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
---+-------------------------------------------
1 | 1
2 | 2 2
3 | 3 3 3
4 | 4 4 6 4
5 | 5 5 15 5 5
6 | 6 6 6 6 15 6
7 | 7 7 7 7 35 7 7
8 | 8 8 12 8 10 12 14 8
9 | 9 9 9 9 45 9 21 18 9
10 | 10 10 15 10 10 15 35 10 45 10
11 | 11 11 33 11 11 33 77 11 99 11 11
12 | 12 12 12 12 15 12 14 12 18 15 33 12
13 | 13 13 13 13 65 13 91 13 39 65 143 13 13
14 | 14 14 14 14 35 14 14 14 21 35 77 14 91 14
PROG
(PARI)
\\ This just returns the least (in a normal sense) number x such that both n and k are in its set of descendants:
up_to = 105;
buildWdescsets(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2, up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1, #f, s = setunion(s, v[n-(n/f[i])])); v[n] = s); (v); }
vdescsets = buildWdescsets(100*up_to); \\ XXX - Think about a safe limit here!
A334231tr(n, k) = for(i=max(n, k), oo, if(setsearch(vdescsets[i], n)&&setsearch(vdescsets[i], k), return(i)));
A334231list(up_to) = { my(v = vector(up_to), i=0); for(n=1, oo, for(k=1, n, i++; if(i > up_to, return(v)); v[i] = A334231tr(n, k))); (v); };
v334231 = A334231list(up_to);
A334231(n) = v334231[n]; \\ Antti Karttunen, Apr 19 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Apr 19 2020
STATUS
approved