A334230 - OEIS

%I #32 Aug 12 2022 19:22:01

%S 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 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 2,3,4,4,6,6,8,6,8,8,12,1,2,3,4,4,6,6,8

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

%H Antti Karttunen, <a href="/A334230/b334230.txt">Table of n, a(n) for n = 1..10440; The first 144 rows, flattened</a>

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

%e 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       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 | 1 2

%e    3 | 1 2 3

%e    4 | 1 2 2 4

%e    5 | 1 2 2 4 5

%e    6 | 1 2 3 4 4 6

%e    7 | 1 2 3 4 4 6 7

%e    8 | 1 2 2 4 4 4 4 8

%e    9 | 1 2 3 4 4 6 6 4 9

%e   10 | 1 2 2 4 5 4 4 8 4 10

%e   11 | 1 2 2 4 5 4 4 8 4 10 11

%e   12 | 1 2 3 4 4 6 6 8 6  8  8 12

%e   13 | 1 2 3 4 4 6 6 8 6  8  8 12 13

%e   14 | 1 2 3 4 4 6 7 8 6  8  8 12 12 14

%o (PARI)

%o \\ This just returns the largest (in a normal sense) number x from the intersection of the set of descendants of n and k:

%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(up_to);

%o A334230tr(n,k) = vecmax(setintersect(vdescsets[n],vdescsets[k]));

%o 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 v334230 = A334230list(up_to);

%o A334230(n) = v334230[n]; \\ _Antti Karttunen_, Apr 19 2020

%Y Cf. A332809, A333123, A334184, A334231.

%K nonn,tabl,look

%O 1,3

%A _Peter Kagey_, _Antti Karttunen_, and _Michael De Vlieger_, Apr 19 2020