A334231 Triangle read by rows: T(n,k) gives the join of n and k in the graded lattice of the positive integers defined by covering relations "n covers (n - n/p)" for all divisors p of n.

1, 2, 2, 3, 3, 3, 4, 4, 6, 4, 5, 5, 15, 5, 5, 6, 6, 6, 6, 15, 6, 7, 7, 7, 7, 35, 7, 7, 8, 8, 12, 8, 10, 12, 14, 8, 9, 9, 9, 9, 45, 9, 21, 18, 9, 10, 10, 15, 10, 10, 15, 35, 10, 45, 10, 11, 11, 33, 11, 11, 33, 77, 11, 99, 11, 11, 12, 12, 12, 12, 15, 12, 14, 12

Offset: 1

Author

Peter Kagey, Apr 19 2020

Keywords

• nonn
• tabl

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.

Examples

			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
		

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

Crossrefs

Cf. A051173, A332809, A333123, A334184, A334230.

Programs

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

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).