

A325612


Width (number of leaves) of the rooted tree with MatulaGoebel number 2^n  1.


4



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
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OFFSET

1,3


COMMENTS

Every positive integer has a unique qfactorization (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 qfactorization of 2^n  1.


LINKS

Table of n, a(n) for n=1..60.
Keith Briggs, Matula numbers and rooted trees.


EXAMPLE

The rooted tree with MatulaGoebel 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^n1], {n, 30}]


CROSSREFS

Cf. A001222, A001221, A056239, A112798.
MatulaGoebel 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,more


AUTHOR

Gus Wiseman, May 12 2019


STATUS

approved



