A350847 Number 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

Offset

1,9

Comments

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

Links

Table of n, a(n) for n=1..87.

Formula

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

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

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

Crossrefs

Positions of first appearances are A001248.

The triangular version is A116482.

Positions of zeros are A346635.

Subtracting from the number of odd conjugate parts gives A350941.

Subtracting from the number of odd parts gives A350942.

Subtracting from the number of even parts gives A350950.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847 (this sequence).

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. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.

Sequence in context: A284585 A280456 A103633 * A026821 A377334 A376971

Adjacent sequences: A350844 A350845 A350846 * A350848 A350849 A350850

Keyword

nonn

Author

Gus Wiseman, Mar 14 2022

Status

approved