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A299757 Weight of the strict integer partition with FDH number n. 36


%S 0,1,2,3,4,3,5,4,6,5,7,5,8,6,6,9,10,7,11,7,7,8,12,6,13,9,8,8,14,7,15,

%T 10,9,11,9,9,16,12,10,8,17,8,18,10,10,13,19,11,20,14,12,11,21,9,11,9,

%U 13,15,22,9,23,16,11,12,12,10,24,13,14,10,25,10,26,17

%N Weight of the strict integer partition with FDH number n.

%C 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. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

%C In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

%e Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

%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 nn=200;FDprimeList=Array[FDfactor,nn,1,Union];

%t FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];

%t Table[Total[FDfactor[n]/.FDrules],{n,nn}]

%Y Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

%K nonn

%O 1,3

%A _Gus Wiseman_, Feb 18 2018

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)