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A325660 Number of ones in the q-signature of n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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)

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.

LINKS

Table of n, a(n) for n=1..87.

MATHEMATICA

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}]

CROSSREFS

Cf. A001694, A055231, A056169, A056239, A112798, A118914, A124010.

Matula-Goebel numbers: A007097, A061775, A109129, A196050, A317713, A324968.

q-factorization: A324922, A324923, A324924, A325614, A325661.

Sequence in context: A209599 A238347 A170942 * A342657 A002187 A124756

Adjacent sequences:  A325657 A325658 A325659 * A325661 A325662 A325663

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 13 2019

STATUS

approved

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Last modified April 10 11:34 EDT 2021. Contains 342845 sequences. (Running on oeis4.)