A324924 Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 4, 7, 1, 1, 1, 2, 2, 1, 1, 1, 8, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 9

Offset

1,3

Comments

Row n is the multiset of Matula-Goebel numbers of all proper terminal subtrees of the rooted tree with Matula-Goebel number n. For example, the rooted tree with Matula-Goebel number 1362 is (o(o)((oo)(oo))), with proper terminal subtrees {o,o,o,o,o,o,(o),(oo),(oo),((oo)(oo))}, which have Matula-Goebel numbers {1,1,1,1,1,1,2,4,4,49}, which is row 1362, as required.

Links

Table of n, a(n) for n=1..83.

Example

Triangle begins:
  {}
  1
  1  2
  1  1
  1  2  3
  1  1  2
  1  1  4
  1  1  1
  1  1  2  2
  1  1  2  3
  1  2  3  5
  1  1  1  2
  1  1  2  6
  1  1  1  4
  1  1  2  2  3
  1  1  1  1
  1  1  4  7
  1  1  1  2  2
  1  1  1  8
  1  1  1  2  3
  1  1  1  2  4
  1  1  2  3  5
  1  1  2  2  9
For example, row 65 is {1,1,1,2,2,3,6} because 65 = q(1)^3 q(2)^2 q(3) q(6).

Mathematica

difac[n_]:=If[n==1, {}, With[{i=PrimePi[FactorInteger[n][[1, 1]]]}, Sort[Prepend[difac[n*i/Prime[i]], i]]]];
Table[difac[n], {n, 30}]

Crossrefs

Cf. A000081, A003963, A061775, A109082, A109129, A112798, A120383, A196050, A317713, A324850, A324922, A324923, A324925, A324931, A324934.

Sequence in context: A000188 A373833 A162401 * A301314 A241898 A191090

Adjacent sequences: A324921 A324922 A324923 * A324925 A324926 A324927

Keyword

nonn,tabf

Author

Gus Wiseman, Mar 20 2019

Status

approved