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A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

%I A350947 #7 Mar 16 2022 16:37:54
%S A350947 1,6,84,210,490,525,2184,2340,5460,9464,12012,12740,12870,13650,14625,
%T A350947 19152,22308,30030,34125,43940,45144,55770,59150,66066,70070,70785,
%U A350947 75075,79625,82992,88920
%N A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.
%C A350947 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.
%F A350947 A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).
%e A350947 The terms together with their prime indices begin:
%e A350947       1: ()
%e A350947       6: (2,1)
%e A350947      84: (4,2,1,1)
%e A350947     210: (4,3,2,1)
%e A350947     490: (4,4,3,1)
%e A350947     525: (4,3,3,2)
%e A350947    2184: (6,4,2,1,1,1)
%e A350947    2340: (6,3,2,2,1,1)
%e A350947    5460: (6,4,3,2,1,1)
%e A350947    9464: (6,6,4,1,1,1)
%e A350947   12012: (6,5,4,2,1,1)
%e A350947   12740: (6,4,4,3,1,1)
%e A350947   12870: (6,5,3,2,2,1)
%e A350947   13650: (6,4,3,3,2,1)
%e A350947   14625: (6,3,3,3,2,2)
%e A350947   19152: (8,4,2,2,1,1,1,1)
%e A350947 For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.
%t A350947 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350947 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350947 Select[Range[1000],Count[primeMS[#],_?EvenQ]==Count[primeMS[#],_?OddQ]==Count[conj[primeMS[#]],_?EvenQ]==Count[conj[primeMS[#]],_?OddQ]&]
%Y A350947 These partitions are counted by A351978.
%Y A350947 There are four individual statistics:
%Y A350947 - A257991 counts odd parts, conjugate A344616.
%Y A350947 - A257992 counts even parts, conjugate A350847.
%Y A350947 There are six possible pairings of statistics:
%Y A350947 - A325698: # of even parts = # of odd parts, counted by A045931.
%Y A350947 - A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350947 - A350848: # of even conj parts = # of odd conj parts, counted by A045931.
%Y A350947 - A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350947 - A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350947 - A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350947 There are three possible double-pairings of statistics:
%Y A350947 - A350946, counted by A351977.
%Y A350947 - A350949, counted by A351976.
%Y A350947 - A351980, counted by A351981.
%Y A350947 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350947 A122111 represents partition conjugation using Heinz numbers.
%Y A350947 A195017 = # of even parts - # of odd parts.
%Y A350947 A316524 = alternating sum of prime indices.
%Y A350947 Cf. A026424, A028260, A098123, A239241, A241638, A325700, A350849, A350941, A350942, A350950, A350951.
%K A350947 nonn
%O A350947 1,2
%A A350947 _Gus Wiseman_, Mar 14 2022