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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 12 2019
STATUS
approved