OFFSET
1,2
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.
LINKS
FORMULA
Number of appearances of n is |a^{-1}(n)| = A004111(n).
EXAMPLE
Sequence of rooted identity trees represented as finitary sets begins:
{}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
{{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
{{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
{{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
{{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 15 2016
STATUS
approved