A355008 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.

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

Offset

1,3

Comments

Conjecture:

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

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

- the other rows converge to some other sequence S:

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

- R and S partition the positive integers,

- for any i, j, k:

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

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

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10011 (rows for n = 1..141 flattened)

Example

Triangle T(n, k) begins:
    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, 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, 9, 11, 13, 14, 16, 18,
    3, 5, 6, 10, 12, 15, 17, 19, 20, 21, 23, 24,
    ...

Prog

(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]) }

Crossrefs

Cf. A135018.

Sequence in context: A193042 A327463 A279478 * A050269 A097151 A306620

Adjacent sequences: A355005 A355006 A355007 * A355009 A355010 A355011

Keyword

nonn,tabl

Author

Rémy Sigrist, Jun 15 2022

Status

approved