A325611 Number of nodes in the rooted tree with Matula-Goebel number 2^n - 1.

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

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)

Then a(n) is one plus the number of factors (counted with multiplicity) 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 12 nodes (o's plus brackets), so a(11) = 12.
		

Crossrefs

Cf. A001222, A001221, A056239, A112798.

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

Mersenne numbers: A046051, A046800, A059305, A325610, A325612, A325625.

Programs

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}]