A334230 Triangle read by rows: T(n,k) gives the meet 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, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 4, 5, 1, 2, 3, 4, 4, 6, 1, 2, 3, 4, 4, 6, 7, 1, 2, 2, 4, 4, 4, 4, 8, 1, 2, 3, 4, 4, 6, 6, 4, 9, 1, 2, 2, 4, 5, 4, 4, 8, 4, 10, 1, 2, 2, 4, 5, 4, 4, 8, 4, 10, 11, 1, 2, 3, 4, 4, 6, 6, 8, 6, 8, 8, 12, 1, 2, 3, 4, 4, 6, 6, 8

Offset: 1

Author

Peter Kagey, Antti Karttunen, and Michael De Vlieger, Apr 19 2020

Keywords

• nonn
• tabl
• look

Comments

Any row with prime index p is a copy of row p-1 followed by that prime p.

Examples

			The interval [1,15] illustrates that, for example, T(12, 10) = 8, T(12, 4) = T(5, 6) = 4, T(8, 3) = 2, 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 | 1 2
   3 | 1 2 3
   4 | 1 2 2 4
   5 | 1 2 2 4 5
   6 | 1 2 3 4 4 6
   7 | 1 2 3 4 4 6 7
   8 | 1 2 2 4 4 4 4 8
   9 | 1 2 3 4 4 6 6 4 9
  10 | 1 2 2 4 5 4 4 8 4 10
  11 | 1 2 2 4 5 4 4 8 4 10 11
  12 | 1 2 3 4 4 6 6 8 6  8  8 12
  13 | 1 2 3 4 4 6 6 8 6  8  8 12 13
  14 | 1 2 3 4 4 6 7 8 6  8  8 12 12 14
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10440; The first 144 rows, flattened
• Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
• Wikipedia, Semilattice

Crossrefs

Cf. A332809, A333123, A334184, A334231.

Programs

PARI

\\ This just returns the largest (in a normal sense) number x from the intersection of the set of descendants of n and k:
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(up_to);
A334230tr(n,k) = vecmax(setintersect(vdescsets[n],vdescsets[k]));
A334230list(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] = A334230tr(n,k))); (v); };
v334230 = A334230list(up_to);
A334230(n) = v334230[n]; \\ Antti Karttunen, Apr 19 2020

Formula

T(n, k) = m*T(n/m, k/m) for m = gcd(n, k).