A325612 Width (number of leaves) of the rooted tree with Matula-Goebel number 2^n - 1.

1, 1, 2, 2, 1, 4, 1, 4, 5, 3, 6, 7, 4, 5, 7, 6, 7, 11, 7, 7, 9, 10, 7, 13, 7, 11, 9, 11, 11, 13, 11, 12, 15, 16, 10, 19, 19, 15, 18, 16, 16, 18, 10, 18, 18, 17, 15, 21, 15, 18, 24, 23, 19, 23, 25, 25, 18, 26, 25, 28, 21, 21, 25, 23, 21, 29, 28, 31, 21, 24, 23

Offset: 1

Author

Gus Wiseman, May 12 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)

For n > 1, a(n) is the multiplicity of q(1) = 2 in the q-factorization of 2^n - 1.

Examples

			The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 6 leaves (o's), so a(11) = 6.
		

Links

• Keith Briggs, Matula numbers and rooted trees.

Crossrefs

Cf. A001222, A001221, A056239, A112798.

Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.

Mersenne numbers: A046051, A046800, A059305, A325610, A325611, A325625.

Programs

Mathematica

mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[2^n-1],{n,30}]

Extensions

More terms from Jinyuan Wang, Feb 25 2025