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A316267 FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.

%I A316267 #5 Jun 29 2018 11:35:32
%S A316267 12,21,57,123,249,417,532,699,867,1100,1389,1463,1509,1708,2049,2068,
%T A316267 2307,2324,2913,3116,3147,3157,3273,3325,3619,3903,4227,4268,4636,
%U A316267 4821,5079,5225,5324,5516,5739,6308,6391,6524,6621,6644,7469,8092,8193,8225,8457
%N A316267 FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.
%C A316267 Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).
%e A316267 Sequence of strict integer partitions of prime numbers with a prime number of prime parts, preceded by their FDH numbers, begins:
%e A316267     12: (3,2)
%e A316267     21: (5,2)
%e A316267     57: (11,2)
%e A316267    123: (17,2)
%e A316267    249: (29,2)
%e A316267    417: (41,2)
%e A316267    532: (11,5,3)
%e A316267    699: (59,2)
%e A316267    867: (71,2)
%e A316267   1100: (13,7,3)
%e A316267   1389: (101,2)
%e A316267   1463: (11,7,5)
%e A316267   1509: (107,2)
%e A316267   1708: (23,5,3)
%t A316267 nn=1000;
%t A316267 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 A316267 FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
%t A316267 Select[Range[nn],And[PrimeQ[Total[FDfactor[#]/.FDrules]],PrimeQ[Length[FDfactor[#]]],And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]
%Y A316267 Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316266.
%K A316267 nonn
%O A316267 1,1
%A A316267 _Gus Wiseman_, Jun 28 2018