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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
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!.
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.
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
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 12 2019
STATUS
approved