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A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1. 4
1, 3, 4, 6, 6, 8, 7, 10, 10, 12, 12, 15, 12, 14, 16, 18, 14, 20, 16, 23, 20, 22, 22, 25, 25, 24, 23, 29, 26, 30, 27, 31, 33, 28, 32, 38, 36, 31, 36, 40, 37, 38, 33, 43, 44, 42, 39, 48, 39, 49, 45, 48, 43, 49, 49, 53, 47, 54, 47, 61 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 one plus the number of factors (counted with multiplicity) in the q-factorization of 2^n - 1.

LINKS

Table of n, a(n) for n=1..60.

EXAMPLE

The rooted tree with Matula-Goebel number 2047 = 2^11 - 1 is (((o)(o))(ooo(o))), which has 12 nodes (o's plus brackets), so a(11) = 12.

MATHEMATICA

mgwt[n_]:=If[n==1, 1, 1+Total[Cases[FactorInteger[n], {p_, k_}:>mgwt[PrimePi[p]]*k]]];

Table[mgwt[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, A325612, A325625.

Sequence in context: A329193 A004219 A077542 * A263842 A286956 A265283

Adjacent sequences:  A325608 A325609 A325610 * A325612 A325613 A325614

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 12 2019

STATUS

approved

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Last modified April 6 04:52 EDT 2020. Contains 333267 sequences. (Running on oeis4.)