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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.

%I A279614 #9 Dec 15 2016 23:36:06
%S A279614 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 A279614 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 A279614 10,10,8,10,9,11,8,9,8,8,9,11,12,9,8,10,10,9
%N 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.
%C A279614 A Fermi-Dirac prime (A050376) is a positive integer of the form p^(2^k) where p is prime and k>=0.
%C A279614 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 A279614 OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%F A279614 Number of appearances of n is |a^{-1}(n)| = A004111(n).
%e A279614 Sequence of rooted identity trees represented as finitary sets begins:
%e A279614 {}, {{}}, {{{}}}, {{{{}}}}, {{{{{}}}}}, {{}{{}}}, {{{{{{}}}}}},
%e A279614 {{}{{{}}}}, {{{}{{}}}}, {{}{{{{}}}}}, {{{{{{{}}}}}}}, {{{}}{{{}}}},
%e A279614 {{{}{{{}}}}}, {{}{{{{{}}}}}}, {{{}}{{{{}}}}}, {{{{}{{}}}}},
%e A279614 {{{}{{{{}}}}}}, {{}{{}{{}}}}, {{{{{{{{}}}}}}}}, {{{{}}}{{{{}}}}},
%e A279614 {{{}}{{{{{}}}}}}, {{}{{{{{{}}}}}}}, {{{{}}{{{}}}}}, {{}{{}}{{{}}}}.
%t A279614 nn=200;
%t A279614 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 A279614 FDprimeList=Array[FDfactor,nn,1,Union];
%t A279614 FDweight[n_?(#<=nn&)]:=If[n===1,1,1+Total[FDweight[Position[FDprimeList,#][[1,1]]]&/@FDfactor[n]]];
%t A279614 Array[FDweight,nn]
%Y A279614 Cf. A004111, A050376, A061773, A061775, A084400, A276625, A279065.
%K A279614 nonn
%O A279614 1,2
%A A279614 _Gus Wiseman_, Dec 15 2016