cp's OEIS Frontend

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

%I A352491 #9 Mar 24 2022 04:10:26
%S A352491 0,0,-1,1,-3,0,-9,3,0,-2,-21,2,-51,-10,-3,9,-111,3,-237,0,-15,-26,
%T A352491 -489,10,-2,-70,2,-12,-995,0,-2017,21,-39,-158,-19,15,-4059,-346,-105,
%U A352491 12,-8151,-18,-16341,-36,-5,-722,-32721,26,-32,5,-237,-108,-65483,19,-53
%N A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.
%C A352491 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 A352491 Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...
%F A352491 a(n) = n - A122111(n).
%e A352491 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 A352491 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A352491 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352491 Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]
%Y A352491 Positions of zeros are A088902, counted by A000700.
%Y A352491 A similar sequence is A175508.
%Y A352491 Positions of nonzero terms are A352486, counted by A330644.
%Y A352491 Positions of negative terms are A352487, counted by A000701.
%Y A352491 Positions of nonnegative terms are A352488, counted by A046682.
%Y A352491 Positions of nonpositive terms are A352489, counted by A046682.
%Y A352491 Positions of positive terms are A352490, counted by A000701.
%Y A352491 A000041 counts integer partitions, strict A000009.
%Y A352491 A003963 is product of prime indices, conjugate A329382.
%Y A352491 A008480 counts permutations of prime indices, conjugate A321648.
%Y A352491 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A352491 A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.
%Y A352491 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352491 A173018 counts permutations by excedances, weak A123125.
%Y A352491 A238744 is partition conjugate of prime signature, ranked by A238745.
%Y A352491 Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.
%K A352491 sign
%O A352491 1,5
%A A352491 _Gus Wiseman_, Mar 20 2022