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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 September 30 19:54 EDT 2023. Contains 365793 sequences. (Running on oeis4.)