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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the combined multiplicity of counts even prime indices of n.

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[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

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[Count[conj[primeMS[n]], _?EvenQ], {n, 100}]

The non-conjugate version is A257992Positions of zeros are A346635.

The Subtracting from the number of odd version is A344616, non-conjugate A257991, reverse A316524parts gives A350941.

Positions Subtracting from the number of zeros appear to be A346635odd parts gives A350942.

The difference with odd conjugate Subtracting from the number of even parts is A350941gives A350950.

There are four statistics:

The difference with - A257991 = # of odd parts is A350942, conjugate A344616.

The difference with - A257992 = # of even parts is A350950, conjugate A350847 (this sequence).

Cf. `A026424, A028260, A130780, A171966, A195017, ~`A236559, ~A236914, `A239241, A241638, A316524, A325700, ~A350841, A350849, A350951.

allocated for Gus WisemanNumber of even parts in the conjugate of the integer partition with Heinz number n.

0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 1, 3, 1, 2, 1, 0, 2, 0, 1, 0, 1, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 0, 4, 1, 0, 0, 2, 1, 0, 2, 3, 1, 2

1,9

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the combined multiplicity of even prime indices of n.

a(n) = A344616(n) - A350941(n).

a(n) = A257992(A122111(n)).

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[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

Positions of first appearances are A001248.

The triangular version is A116482.

The non-conjugate version is A257992.

The odd version is A344616, non-conjugate A257991, reverse A316524.

Positions of zeros appear to be A346635.

The difference with odd conjugate parts is A350941.

The difference with odd parts is A350942.

The difference with even parts is A350950.

There are six possible pairings of statistics:

- A325698: # of even parts = # of odd parts, counted by A045931.

- A349157: # of even parts = # of odd conjugate parts, counted by A277579.

- A350848: # of even conj parts = # of odd conj parts, counted by A045931.

- A350943: # of even conjugate parts = # of odd parts, counted by A277579.

- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

- A350945: # of even parts = # of even conjugate parts, counted by A350948.

There are three possible double-pairings of statistics:

- A350946, counted by A351977.

- A350949, counted by A351976.

- A351980, counted by A351981.

The case of all four statistics equal is A350947, counted by A351978.

A056239 adds up prime indices, counted by A001222, row sums of A112798.

A122111 represents partition conjugation using Heinz numbers.

Cf. `A026424, A028260, A130780, A171966, A195017, ~`A236559, ~A236914, `A239241, A241638, A325700, ~A350841, A350849, A350951.

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Gus Wiseman, Mar 14 2022

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editing

allocated for Gus Wiseman

allocated

approved