%I #10 Oct 04 2018 10:15:32

%S 0,1,2,1,3,2,4,1,4,3,5,2,6,4,5,1,7,4,8,3,6,5,9,2,6,6,6,4,10,5,11,1,7,

%T 7,7,4,12,8,8,3,13,6,14,5,7,9,15,2,8,6,9,6,16,6,8,4,10,10,17,5,18,11,

%U 8,1,9,7,19,7,11,7,20,4,21,12,8,8,9,8,22,3,8

%N Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%F a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - _Charlie Neder_, Oct 03 2018

%e a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5.

%t ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];

%t nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];

%t Table[Min[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

%Y Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856.

%K nonn

%O 1,3

%A _Gus Wiseman_, Sep 29 2018