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n minus the Heinz number of the conjugate of the integer partition with Heinz number n.
17

%I #9 Mar 24 2022 04:10:26

%S 0,0,-1,1,-3,0,-9,3,0,-2,-21,2,-51,-10,-3,9,-111,3,-237,0,-15,-26,

%T -489,10,-2,-70,2,-12,-995,0,-2017,21,-39,-158,-19,15,-4059,-346,-105,

%U 12,-8151,-18,-16341,-36,-5,-722,-32721,26,-32,5,-237,-108,-65483,19,-53

%N n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

%F a(n) = n - A122111(n).

%e The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

%Y Positions of zeros are A088902, counted by A000700.

%Y A similar sequence is A175508.

%Y Positions of nonzero terms are A352486, counted by A330644.

%Y Positions of negative terms are A352487, counted by A000701.

%Y Positions of nonnegative terms are A352488, counted by A046682.

%Y Positions of nonpositive terms are A352489, counted by A046682.

%Y Positions of positive terms are A352490, counted by A000701.

%Y A000041 counts integer partitions, strict A000009.

%Y A003963 is product of prime indices, conjugate A329382.

%Y A008480 counts permutations of prime indices, conjugate A321648.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.

%Y A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

%Y A173018 counts permutations by excedances, weak A123125.

%Y A238744 is partition conjugate of prime signature, ranked by A238745.

%Y Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.

%K sign

%O 1,5

%A _Gus Wiseman_, Mar 20 2022