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A325609 Unsorted q-signature of n!. Irregular triangle read by rows where T(n,k) is the multiplicity of q(k) in the factorization of n! into factors q(i) = prime(i)/i. 3
1, 2, 1, 4, 1, 5, 2, 1, 7, 3, 1, 9, 3, 1, 1, 12, 3, 1, 1, 14, 5, 1, 1, 16, 6, 2, 1, 17, 7, 3, 1, 1, 20, 8, 3, 1, 1, 22, 9, 3, 1, 1, 1, 25, 9, 3, 2, 1, 1, 27, 11, 4, 2, 1, 1, 31, 11, 4, 2, 1, 1, 33, 11, 4, 3, 1, 1, 1, 36, 13, 4, 3, 1, 1, 1, 39, 13, 4, 3, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 is the sequence of nonzero exponents in the q-factorization of n!.

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..83.

EXAMPLE

We have 10! = q(1)^16 q(2)^6 q(3)^2 q(4), so row n = 10 is (16,6,2,1).

Triangle begins:

  {}

   1

   2  1

   4  1

   5  2  1

   7  3  1

   9  3  1  1

  12  3  1  1

  14  5  1  1

  16  6  2  1

  17  7  3  1  1

  20  8  3  1  1

  22  9  3  1  1  1

  25  9  3  2  1  1

  27 11  4  2  1  1

  31 11  4  2  1  1

  33 11  4  3  1  1  1

  36 13  4  3  1  1  1

  39 13  4  3  1  1  1  1

  42 14  5  3  1  1  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, 20}]

CROSSREFS

Row lengths are A000720.

Row sums are A325544(n) - 1.

Column k = 1 is A325543.

Cf. A056239, A067255, A112798, A118914, A124010.

Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324935.

Factorial numbers: A000142, A011371, A022559, A071626, A115627, A325276.

q-factorization: A324922, A324923, A324924, A325614, A325615, A325660.

Sequence in context: A088296 A282738 A093890 * A006306 A322100 A277100

Adjacent sequences:  A325606 A325607 A325608 * A325610 A325611 A325612

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, May 12 2019

STATUS

approved

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Last modified December 13 01:23 EST 2019. Contains 329963 sequences. (Running on oeis4.)