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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.

%I A334230 #32 Aug 12 2022 19:22:01
%S A334230 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,
%T A334230 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,
%U A334230 2,3,4,4,6,6,8,6,8,8,12,1,2,3,4,4,6,6,8
%N 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.
%C A334230 Any row with prime index p is a copy of row p-1 followed by that prime p.
%H A334230 Antti Karttunen, <a href="/A334230/b334230.txt">Table of n, a(n) for n = 1..10440; The first 144 rows, flattened</a>
%H A334230 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/a/3640072/121988">Does a graded poset on the positive integers generated from subtracting factors define a lattice?</a>
%H A334230 Wikipedia, <a href="https://en.wikipedia.org/wiki/Semilattice">Semilattice</a>
%F A334230 T(n, k) = m*T(n/m, k/m) for m = gcd(n, k).
%e A334230 The interval [1,15] illustrates that, for example, T(12, 10) = 8, T(12, 4) = T(5, 6) = 4, T(8, 3) = 2, etc.
%e A334230       15
%e A334230      _/ \_
%e A334230     /     \
%e A334230   10       12
%e A334230   | \_   _/ |
%e A334230   |   \ /   |
%e A334230   5    8    6
%e A334230    \_  |  _/|
%e A334230      \_|_/  |
%e A334230        4    3
%e A334230        |  _/
%e A334230        |_/
%e A334230        2
%e A334230        |
%e A334230        |
%e A334230        1
%e A334230 Triangle begins:
%e A334230   n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e A334230   ---+---------------------------------
%e A334230    1 | 1
%e A334230    2 | 1 2
%e A334230    3 | 1 2 3
%e A334230    4 | 1 2 2 4
%e A334230    5 | 1 2 2 4 5
%e A334230    6 | 1 2 3 4 4 6
%e A334230    7 | 1 2 3 4 4 6 7
%e A334230    8 | 1 2 2 4 4 4 4 8
%e A334230    9 | 1 2 3 4 4 6 6 4 9
%e A334230   10 | 1 2 2 4 5 4 4 8 4 10
%e A334230   11 | 1 2 2 4 5 4 4 8 4 10 11
%e A334230   12 | 1 2 3 4 4 6 6 8 6  8  8 12
%e A334230   13 | 1 2 3 4 4 6 6 8 6  8  8 12 13
%e A334230   14 | 1 2 3 4 4 6 7 8 6  8  8 12 12 14
%o A334230 (PARI)
%o A334230 \\ This just returns the largest (in a normal sense) number x from the intersection of the set of descendants of n and k:
%o A334230 up_to = 105;
%o A334230 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); }
%o A334230 vdescsets = buildWdescsets(up_to);
%o A334230 A334230tr(n,k) = vecmax(setintersect(vdescsets[n],vdescsets[k]));
%o A334230 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); };
%o A334230 v334230 = A334230list(up_to);
%o A334230 A334230(n) = v334230[n]; \\ _Antti Karttunen_, Apr 19 2020
%Y A334230 Cf. A332809, A333123, A334184, A334231.
%K A334230 nonn,tabl,look
%O A334230 1,3
%A A334230 _Peter Kagey_, _Antti Karttunen_, and _Michael De Vlieger_, Apr 19 2020