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%I #7 Mar 16 2022 16:38:01
%S 1,6,65,84,210,216,319,490,525,532,731,1254,1403,1924,2184,2340,2449,
%T 2470,3024,3135,3325,3774,4028,4141,4522,5311,5460,7030,7314,7315,
%U 7560,7776,7942,8201,8236,9048,9435,9464,10659,10921,11484,11914,12012,12025,12740
%N Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate 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 Closed under A122111 (conjugation).
%F Intersection of A325698 and A350848.
%F A257992(a(n)) = A257991(a(n)).
%F A350847(a(n)) = A344616(a(n)).
%e The terms together with their prime indices begin:
%e 1: ()
%e 6: (2,1)
%e 65: (6,3)
%e 84: (4,2,1,1)
%e 210: (4,3,2,1)
%e 216: (2,2,2,1,1,1)
%e 319: (10,5)
%e 490: (4,4,3,1)
%e 525: (4,3,3,2)
%e 532: (8,4,1,1)
%e 731: (14,7)
%e 1254: (8,5,2,1)
%e 1403: (18,9)
%e 1924: (12,6,1,1)
%e 2184: (6,4,2,1,1,1)
%e 2340: (6,3,2,2,1,1)
%e 2449: (22,11)
%e 2470: (8,6,3,1)
%e For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.
%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],#==1||Mean[Mod[primeMS[#],2]]== Mean[Mod[conj[primeMS[#]],2]]==1/2&]
%Y For the first condition alone:
%Y - counted by A045931 (strict A239241)
%Y - ordered version (compositions) A098123
%Y - ranked by A325698
%Y - without multiplicity A325700 (counted by A241638)
%Y The second condition alone is ranked by A350848, strict A352129.
%Y These partitions are counted by A351977, strict A352128.
%Y There are four statistics:
%Y - A257991 = # of odd parts, conjugate A344616.
%Y - A257992 = # of even parts, conjugate A350847.
%Y There are four other possible pairings of statistics:
%Y - A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y - A350943: # of even conj parts = # of odd parts, strict counted by A352130.
%Y - A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y - A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y There are two other possible double-pairings of statistics:
%Y - A350949, counted by A351976.
%Y - A351980, counted by A351981.
%Y The case of all four statistics equal is A350947, counted by A351978.
%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 A316524 = alternating sum of prime indices.
%Y Cf. A026424, A028260, A130780, A171966, A347450, A350849, A350941, A350942, A350950, A350951.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 14 2022
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