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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
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.
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: A154844 A351089 A133831 * A305054 A375148 A238097
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 12 2019
STATUS
approved