cp's OEIS Frontend

A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).

%I A305830 #5 Jun 11 2018 13:12:59
%S A305830 0,0,1,1,1,1,1,1,2,1,1,2,2,1,2,1,2,2,1,2,2,1,2,2,1,2,3,2,2,2,2,1,2,2,
%T A305830 2,3,1,1,3,2,1,2,2,2,3,2,1,2,2,1,3,3,2,3,2,2,2,2,2,3,1,2,3,2,3,2,3,3,
%U A305830 3,2,1,3,2,1,2,2,2,3,2,2,2,1,1,3,3,2,3
%N A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).
%C A305830 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) = w(s_1) + ... + w(s_k) where w = A064547.
%e A305830 Sequence of FDH set-systems (a list containing all finite sets of finite sets of positive integers) begins:
%e A305830    1: {}
%e A305830    2: {{}}
%e A305830    3: {{1}}
%e A305830    4: {{2}}
%e A305830    5: {{3}}
%e A305830    6: {{},{1}}
%e A305830    7: {{4}}
%e A305830    8: {{},{2}}
%e A305830    9: {{1,2}}
%e A305830   10: {{},{3}}
%e A305830   11: {{5}}
%e A305830   12: {{1},{2}}
%e A305830   13: {{1,3}}
%e A305830   14: {{},{4}}
%e A305830   15: {{1},{3}}
%e A305830   16: {{6}}
%e A305830   17: {{1,4}}
%e A305830   18: {{},{1,2}}
%e A305830   19: {{7}}
%e A305830   20: {{2},{3}}
%e A305830   21: {{1},{4}}
%e A305830   22: {{},{5}}
%e A305830   23: {{2,3}}
%e A305830   24: {{},{1},{2}}
%e A305830   25: {{8}}
%e A305830   26: {{},{1,3}}
%e A305830   27: {{1},{1,2}}
%t A305830 nn=100;
%t A305830 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 A305830 FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t A305830 Table[Total[Length/@(FDfactor/@(FDfactor[n]/.FDrules))],{n,nn}]
%Y A305830 Cf. A003963, A050376, A056239, A061775, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A302242, A302243, A305829, A305832.
%K A305830 nonn
%O A305830 1,9
%A A305830 _Gus Wiseman_, Jun 10 2018