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A319911 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.

%I A319911 #5 Oct 01 2018 21:17:15
%S A319911 0,1,1,2,3,7,9,21,31,65,102,194,321,575,956,1652,2684,4576,7367,12035,
%T A319911 19490,31185,49418,78595,123393
%N A319911 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.
%e A319911 The a(6) = 7 pairs:
%e A319911   6 <= (6)
%e A319911   6 <= (4,2)
%e A319911   8 <= (4,2)
%e A319911   6 <= (3,3)
%e A319911   9 <= (3,3)
%e A319911   6 <= (2,2,2)
%e A319911   8 <= (2,2,2)
%e A319911 The a(7) = 9 pairs:
%e A319911    7 <= (7)
%e A319911    7 <= (5,2)
%e A319911   10 <= (5,2)
%e A319911    7 <= (4,3)
%e A319911   12 <= (4,3)
%e A319911    7 <= (3,2,2)
%e A319911    8 <= (3,2,2)
%e A319911   10 <= (3,2,2)
%e A319911   12 <= (3,2,2)
%t A319911 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A319911 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 A319911 Table[Total[Length/@nexos/@Select[IntegerPartitions[n],FreeQ[#,1]&]],{n,30}]
%Y A319911 Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319912, A319913.
%K A319911 nonn,more
%O A319911 1,4
%A A319911 _Gus Wiseman_, Oct 01 2018