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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 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.)