A325613 Full q-signature of n. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the q-factorization of n.

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 0, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 3, 1, 2, 1, 0, 0, 0, 1, 3, 0, 0, 1, 2, 2, 1, 4, 2, 0, 0, 1, 0, 0, 1, 3, 2, 3, 0, 0, 0, 0, 0, 0, 1, 3, 1, 1, 3, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 1, 4, 1, 2, 2, 2, 3, 1, 0, 0

Offset

1,4

Comments

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:

11 = q(1) q(2) q(3) q(5)

50 = q(1)^3 q(2)^2 q(3)^2

360 = q(1)^6 q(2)^3 q(3)

Also the number of terminal subtrees with Matula-Goebel number k of the rooted tree with Matula-Goebel number n.

Links

Table of n, a(n) for n=1..87.

Example

Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 0 0 1
  3
  2 2
  2 1 1
  1 1 1 0 1
  3 1
  2 1 0 0 0 1
  3 0 0 1
  2 2 1
  4
  2 0 0 1 0 0 1
  3 2
  3 0 0 0 0 0 0 1
  3 1 1

Mathematica

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

Crossrefs

Row lengths are A061395.

Row sums are A196050.

Row-maxima are A109129.

The number whose full prime signature is the n-th row is A324922(n).

Cf. A067255.

Matula-Goebel numbers: A007097, A061775, A109082, A317713.

q-factorization: A324923, A324924, A325613, A325614, A325615, A325660.

Sequence in context: A154844 A351089 A133831 * A305054 A238097 A066955

Adjacent sequences: A325610 A325611 A325612 * A325614 A325615 A325616

Keyword

nonn,tabf

Author

Gus Wiseman, May 12 2019

Status

approved