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A325660
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Number of ones in the q-signature of n.
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6
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0, 1, 2, 0, 3, 1, 1, 0, 0, 2, 4, 1, 2, 1, 1, 0, 2, 0, 1, 2, 2, 3, 1, 1, 0, 2, 0, 1, 3, 1, 5, 0, 2, 2, 3, 0, 2, 1, 1, 2, 3, 2, 2, 3, 1, 1, 2, 1, 0, 0, 3, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 3, 3, 0, 3, 2, 0, 1, 4, 1, 4, 2, 0, 3, 2, 2, 4, 2, 2
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OFFSET
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1,3
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COMMENTS
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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)
Then a(n) is the number of factors of multiplicity one in the q-factorization of n.
Also the number of rooted trees appearing only once in the multiset of terminal subtrees of the rooted tree with Matula-Goebel number n.
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LINKS
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MATHEMATICA
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difac[n_]:=If[n==1, {}, With[{i=PrimePi[FactorInteger[n][[1, 1]]]}, Sort[Prepend[difac[n*i/Prime[i]], i]]]];
Table[Count[Length/@Split[difac[n]], 1], {n, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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