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A325614 Unsorted q-signature of n. 7
1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 4, 2, 1, 1, 3, 2, 3, 1, 3, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 2, 1, 4, 1, 2, 2, 2, 3, 1, 1, 3, 3, 4, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 3, 1, 1, 3, 1, 1 (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)

Row n lists the nonzero multiplicities in the q-factorization of n, in order of q-index. For example, row 11 is (1,1,1,1) and row 360 is (6,3,1).

LINKS

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

EXAMPLE

Triangle begins:

  {}

  1

  1 1

  2

  1 1 1

  2 1

  2 1

  3

  2 2

  2 1 1

  1 1 1 1

  3 1

  2 1 1

  3 1

  2 2 1

  4

  2 1 1

  3 2

  3 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]]]];

Table[Length/@Split[difac[n]], {n, 30}]

CROSSREFS

Row lengths are A324923.

Row sums are A196050.

Row-maxima are A109129.

Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325615, A325660.

Sequence in context: A136755 A156775 A064693 * A319608 A230850 A072085

Adjacent sequences:  A325611 A325612 A325613 * A325615 A325616 A325617

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, May 12 2019

STATUS

approved

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Last modified April 11 18:00 EDT 2021. Contains 342888 sequences. (Running on oeis4.)