A279614 a(1)=1, a(d(x_1)*..*d(x_k)) = 1+a(x_1)+..+a(x_k) where d(n) = n-th Fermi-Dirac prime.

1, 2, 3, 4, 5, 4, 6, 5, 5, 6, 7, 6, 6, 7, 7, 6, 7, 6, 8, 8, 8, 8, 7, 7, 7, 7, 7, 9, 8, 8, 8, 7, 9, 8, 10, 8, 7, 9, 8, 9, 8, 9, 7, 10, 9, 8, 9, 8, 9, 8, 9, 9, 9, 8, 11, 10, 10, 9, 9, 10, 8, 9, 10, 9, 10, 10, 8, 10, 9, 11, 8, 9, 8, 8, 9, 11, 12, 9, 8, 10, 10, 9

Offset: 1

Author

Gus Wiseman, Dec 15 2016

Keywords

• nonn

Comments

A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.

In analogy with the Matula-Goebel correspondence between rooted trees and positive integers (see A061775), the iterated normalized Fermi-Dirac representation gives a correspondence between rooted identity trees and positive integers. Then a(n) is the number of nodes in the rooted identity tree corresponding to n.

Examples

			Sequence of rooted identity trees represented as finitary sets begins:
{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
{{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
{{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
{{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
{{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.
		

Links

• OEIS Wiki, "Fermi-Dirac representation" of n

Crossrefs

Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065.

Programs

Mathematica

nn=200;
FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];
FDweight[n_?(#<=nn&)]:=If[n===1,1,1+Total[FDweight[Position[FDprimeList,#][[1,1]]]&/@FDfactor[n]]];
Array[FDweight,nn]

Formula

Number of appearances of n is |a^{-1}(n)| = A004111(n).