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