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%I #11 May 20 2022 09:27:06
%S 1,2,6,9,20,30,56,75,84,125,176,210,264,294,315,350,416,441,490,525,
%T 624,660,735,924,990,1088,1100,1386,1540,1560,1632,1650,1715,2184,
%U 2310,2340,2401,2432,2600,3267,3276,3388,3640,3648,3900,4080,4125,5082,5324,5390
%N Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.
%C 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 Intersection of A350944 and A350945.
%F A257991(a(n)) = A344616(a(n)).
%F A257992(a(n)) = A350847(a(n)).
%F Closed under A122111 (conjugation).
%e The terms together with their prime indices begin:
%e 1: ()
%e 2: (1)
%e 6: (2,1)
%e 9: (2,2)
%e 20: (3,1,1)
%e 30: (3,2,1)
%e 56: (4,1,1,1)
%e 75: (3,3,2)
%e 84: (4,2,1,1)
%e 125: (3,3,3)
%e 176: (5,1,1,1,1)
%e 210: (4,3,2,1)
%e 264: (5,2,1,1,1)
%e 294: (4,4,2,1)
%e 315: (4,3,2,2)
%e 350: (4,3,3,1)
%e 416: (6,1,1,1,1,1)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t Select[Range[1000],Count[primeMS[#],_?OddQ]==Count[conj[primeMS[#]],_?OddQ]&&Count[primeMS[#],_?EvenQ]==Count[conj[primeMS[#]],_?EvenQ]&]
%Y The second condition alone is A350944, counted by A277103.
%Y The first condition alone is A350945, counted by A350948.
%Y The case of all four statistics equal is A350947, counted by A351978.
%Y These partitions are counted by A351976.
%Y There are four other possible pairings of statistics:
%Y - A045931: # even = # odd, ranked by A325698, strict A239241.
%Y - A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
%Y - A277579: # even = # odd conj, ranked by A349157, strict A352131.
%Y - A277579: # even conj = # odd, ranked by A350943, strict A352130.
%Y There are two other possible double-pairings of statistics:
%Y - A350946, counted by A351977.
%Y - A351980, counted by A351981.
%Y A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A122111 represents partition conjugation using Heinz numbers.
%Y A195017 = # of even parts - # of odd parts.
%Y A257991 counts odd parts, conjugate A344616.
%Y A257992 counts even parts, conjugate A350847.
%Y A316524 = alternating sum of prime indices.
%Y Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350849, A350941, A350942, A350950, A350951.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 14 2022
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