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A325615
Sorted q-signature of n.
5
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 3, 3, 1, 4, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1
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)
Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).
EXAMPLE
Triangle begins:
{}
1
1 1
2
1 1 1
1 2
1 2
3
2 2
1 1 2
1 1 1 1
1 3
1 1 2
1 3
1 2 2
4
1 1 2
2 3
1 3
1 1 3
MATHEMATICA
difac[n_]:=If[n==1, {}, With[{i=PrimePi[FactorInteger[n][[1, 1]]]}, Sort[Prepend[difac[n*i/Prime[i]], i]]]];
Table[Sort[Length/@Split[difac[n]]], {n, 30}]
CROSSREFS
Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Sequence in context: A327659 A327519 A287917 * A029434 A358192 A156281
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 12 2019
STATUS
approved