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%I #9 Dec 15 2016 23:36:06
%S 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,
%T 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,
%U 10,10,8,10,9,11,8,9,8,8,9,11,12,9,8,10,10,9
%N 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.
%C A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.
%C 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.
%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%F Number of appearances of n is |a^{-1}(n)| = A004111(n).
%e Sequence of rooted identity trees represented as finitary sets begins:
%e {}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
%e {{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
%e {{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
%e {{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
%e {{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.
%t nn=200;
%t 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)]]]]];
%t FDprimeList=Array[FDfactor,nn,1,Union];
%t FDweight[n_?(#<=nn&)]:=If[n===1,1,1+Total[FDweight[Position[FDprimeList,#][[1,1]]]&/@FDfactor[n]]];
%t Array[FDweight,nn]
%Y Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065.
%K nonn
%O 1,2
%A _Gus Wiseman_, Dec 15 2016
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