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

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Author

Gus Wiseman, Mar 20 2022

Keywords

• sign

Comments

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.

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

Examples

			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.
		

Crossrefs

Positions of zeros are A088902, counted by A000700.

A similar sequence is A175508.

Positions of nonzero terms are A352486, counted by A330644.

Positions of negative terms are A352487, counted by A000701.

Positions of nonnegative terms are A352488, counted by A046682.

Positions of nonpositive terms are A352489, counted by A046682.

Positions of positive terms are A352490, counted by A000701.

A000041 counts integer partitions, strict A000009.

A003963 is product of prime indices, conjugate A329382.

A008480 counts permutations of prime indices, conjugate A321648.

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

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

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

A173018 counts permutations by excedances, weak A123125.

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

Formula

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