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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A290084 A154844 A133831 * A305054 A238097 A066955

Adjacent sequences:  A325610 A325611 A325612 * A325614 A325615 A325616

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, May 12 2019

STATUS

approved

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)