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

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

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.

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.

Programs

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}]