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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.

%I A334231 #37 Aug 12 2022 19:21:29
%S A334231 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,
%T A334231 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,
%U A334231 33,11,11,33,77,11,99,11,11,12,12,12,12,15,12,14,12
%N 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.
%C A334231 The poset of the positive integers is defined by covering relations "n covers (n - n/p)" for all divisors p of n.
%C A334231 n appears A332809(n) times in row n.
%H A334231 Peter Kagey, <a href="/A334231/b334231.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%H A334231 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 A334231 Wikipedia, <a href="https://en.wikipedia.org/wiki/Semilattice">Semilattice</a>
%F A334231 T(n,1) = T(n,n) = n. T(n, 2) = n for n >= 2.
%F A334231 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).
%F A334231 Moreover, empirically it looks like T(x,y) divides lcm(x,y).
%e A334231 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.
%e A334231       15
%e A334231      _/ \_
%e A334231     /     \
%e A334231   10       12
%e A334231   | \_   _/ |
%e A334231   |   \ /   |
%e A334231   5    8    6
%e A334231    \_  |  _/|
%e A334231      \_|_/  |
%e A334231        4    3
%e A334231        |  _/
%e A334231        |_/
%e A334231        2
%e A334231        |
%e A334231        |
%e A334231        1
%e A334231 Triangle begins:
%e A334231   n\k|  1  2  3  4  5  6  7  8  9 10  11 12 13 14
%e A334231   ---+-------------------------------------------
%e A334231    1 |  1
%e A334231    2 |  2  2
%e A334231    3 |  3  3  3
%e A334231    4 |  4  4  6  4
%e A334231    5 |  5  5 15  5  5
%e A334231    6 |  6  6  6  6 15  6
%e A334231    7 |  7  7  7  7 35  7  7
%e A334231    8 |  8  8 12  8 10 12 14  8
%e A334231    9 |  9  9  9  9 45  9 21 18  9
%e A334231   10 | 10 10 15 10 10 15 35 10 45 10
%e A334231   11 | 11 11 33 11 11 33 77 11 99 11  11
%e A334231   12 | 12 12 12 12 15 12 14 12 18 15  33 12
%e A334231   13 | 13 13 13 13 65 13 91 13 39 65 143 13 13
%e A334231   14 | 14 14 14 14 35 14 14 14 21 35  77 14 91 14
%o A334231 (PARI)
%o A334231 \\ This just returns the least (in a normal sense) number x such that both n and k are in its set of descendants:
%o A334231 up_to = 105;
%o A334231 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 A334231 vdescsets = buildWdescsets(100*up_to); \\ XXX - Think about a safe limit here!
%o A334231 A334231tr(n,k) = for(i=max(n,k),oo,if(setsearch(vdescsets[i],n)&&setsearch(vdescsets[i],k),return(i)));
%o A334231 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); };
%o A334231 v334231 = A334231list(up_to);
%o A334231 A334231(n) = v334231[n]; \\ _Antti Karttunen_, Apr 19 2020
%Y A334231 Cf. A051173, A332809, A333123, A334184, A334230.
%K A334231 nonn,tabl
%O A334231 1,2
%A A334231 _Peter Kagey_, Apr 19 2020