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

1,3

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)

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

Links

Table of n, a(n) for n=1..71.

Keith Briggs, Matula numbers and rooted trees.

Example

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.

Mathematica

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

Crossrefs

Cf. A001222, A001221, A056239, A112798.

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

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

Sequence in context: A129721 A268193 A238606 * A054995 A018219 A174714

Adjacent sequences: A325609 A325610 A325611 * A325613 A325614 A325615

Keyword

nonn

Author

Gus Wiseman, May 12 2019

Extensions

More terms from Jinyuan Wang, Feb 25 2025

Status

approved