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A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.

%I A305829 #13 Aug 04 2018 06:30:18
%S A305829 1,1,2,3,4,2,5,3,6,4,7,6,8,5,8,9,10,6,11,12,10,7,12,6,13,8,12,15,14,8,
%T A305829 15,9,14,10,20,18,16,11,16,12,17,10,18,21,24,12,19,18,20,13,20,24,21,
%U A305829 12,28,15,22,14,22,24,23,15,30,27,32,14,24,30,24,20,25
%N A305829 Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then multiply everything together.
%C A305829 Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = s_1 * ... * s_k.
%C A305829 Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - _Andrew Howroyd_, Aug 02 2018
%H A305829 Andrew Howroyd, <a href="/A305829/b305829.txt">Table of n, a(n) for n = 1..1000</a>
%H A305829 Gus Wiseman, <a href="/A305829/a305829.png">Tree of x -> a(x) for n = 1...75</a>
%t A305829 nn=100;
%t A305829 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 A305829 FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t A305829 Table[Times@@(FDfactor[n]/.FDrules),{n,nn}]
%o A305829 (PARI) \\ here isfd is membership test for A050376.
%o A305829 isfd(n)={my(e=isprimepower(n)); e && e == 1<<valuation(e, 2)}
%o A305829 seq(n)={my(v=select(isfd, [1..n])); vector(n, n, my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); prod(j=0, logint(e,2), if(bittest(e,j), vecsearch(v, p^(1<<j)), 1))))} \\ _Andrew Howroyd_, Aug 02 2018
%Y A305829 Cf. A003963, A050376, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A305830, A305831, A305832.
%K A305829 nonn,mult
%O A305829 1,3
%A A305829 _Gus Wiseman_, Jun 10 2018