A355008 - OEIS

%I #18 Jun 18 2022 08:42:46

%S 1,1,2,3,4,5,1,2,7,9,1,2,7,9,11,3,5,6,8,10,12,1,2,4,7,9,11,13,1,2,4,7,

%T 9,11,13,14,3,5,6,10,12,15,17,19,20,1,2,4,7,8,9,11,13,14,16,1,2,4,7,8,

%U 9,11,13,14,16,18,3,5,6,10,12,15,17,19,20,21,23,24

%N Triangle T(n, k), n > 0, k = 1..n, read by rows; each row contains the smallest distinct positive integers (in ascending order) such that a term of row n times a term of row n+1 does not appear in row n+2.

%C Conjecture:

%C - rows with indexes of the form 3*k converge to some sequence R:

%C     R = 3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24, 27, 29, 30, 33, 34, ...

%C - the other rows converge to some other sequence S:

%C     S = 1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 18, 22, 25, 26, 28, 31, 32, 36, ...

%C - R and S partition the positive integers,

%C - for any i, j, k:

%C     R(i) <> S(j) * S(k),

%C     S(i) <> R(k) * S(k).

%H Rémy Sigrist, <a href="/A355008/b355008.txt">Table of n, a(n) for n = 1..10011</a> (rows for n = 1..141 flattened)

%e Triangle T(n, k) begins:

%e     1,

%e     1, 2,

%e     3, 4, 5,

%e     1, 2, 7, 9,

%e     1, 2, 7, 9, 11,

%e     3, 5, 6, 8, 10, 12,

%e     1, 2, 4, 7, 9, 11, 13,

%e     1, 2, 4, 7, 9, 11, 13, 14,

%e     3, 5, 6, 10, 12, 15, 17, 19, 20,

%e     1, 2, 4, 7, 8, 9, 11, 13, 14, 16,

%e     1, 2, 4, 7, 8, 9, 11, 13, 14, 16, 18,

%e     3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24,

%e     ...

%o (PARI) { rrr=rr=[]; for (n=1, 12, x=setbinop((i,j)->i*j, rrr,rr); r=vector(n); k=0; for (v=1, oo, if (!setsearch(x,v), print1 (r[k++]=v", "); if (k==n, break))); [rrr,rr]=[rr,r]) }

%Y Cf. A135018.

%K nonn,tabl

%O 1,3

%A _Rémy Sigrist_, Jun 15 2022