A325660 Number of ones in the q-signature of n.

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

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: A263322 A353496 A170942 * A342657 A002187 A124756

Adjacent sequences: A325657 A325658 A325659 * A325661 A325662 A325663

Keyword

nonn

Author

Gus Wiseman, May 13 2019

Status

approved