cp's OEIS Frontend

A351980 Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.

%I A351980 #9 May 20 2022 09:27:00
%S A351980 1,6,84,126,140,210,490,525,686,875,1404,1456,2106,2184,2288,2340,
%T A351980 3432,3510,5460,6760,7644,8190,8580,8775,9100,9464,11466,12012,12740,
%U A351980 12870,13650,14300,14625,15808,18018,18468,19110,19152,20020,20672,21450,22308,23712
%N A351980 Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.
%C A351980 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 A351980 Closed under A122111 (conjugation).
%F A351980 Intersection of A349157 and A350943.
%F A351980 A257992(a(n)) = A344616(a(n)).
%F A351980 A257991(a(n)) = A350847(a(n)).
%e A351980 The terms together with their prime indices begin:
%e A351980      1: ()
%e A351980      6: (2,1)
%e A351980     84: (4,2,1,1)
%e A351980    126: (4,2,2,1)
%e A351980    140: (4,3,1,1)
%e A351980    210: (4,3,2,1)
%e A351980    490: (4,4,3,1)
%e A351980    525: (4,3,3,2)
%e A351980    686: (4,4,4,1)
%e A351980    875: (4,3,3,3)
%e A351980   1404: (6,2,2,2,1,1)
%e A351980   1456: (6,4,1,1,1,1)
%e A351980   2106: (6,2,2,2,2,1)
%e A351980   2184: (6,4,2,1,1,1)
%e A351980   2288: (6,5,1,1,1,1)
%e A351980   2340: (6,3,2,2,1,1)
%t A351980 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A351980 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A351980 Select[Range[1000],Count[primeMS[#],_?EvenQ]==Count[conj[primeMS[#]],_?OddQ]&&Count[primeMS[#],_?OddQ]==Count[conj[primeMS[#]],_?EvenQ]&]
%Y A351980 The first condition alone is A349157, counted by A277579.
%Y A351980 The second condition alone is A350943, counted by A277579.
%Y A351980 There are two other possible double-pairings of statistics:
%Y A351980 - A350946, counted by A351977.
%Y A351980 - A350949, counted by A351976.
%Y A351980 The case of all four statistics equal is A350947, counted by A351978.
%Y A351980 These partitions are counted by A351981.
%Y A351980 Partitions with as many even as odd parts:
%Y A351980 - counted by A045931
%Y A351980 - strict case counted by A239241
%Y A351980 - ranked by A325698
%Y A351980 - conjugate ranked by A350848
%Y A351980 - strict conjugate case counted by A352129
%Y A351980 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A351980 A122111 represents partition conjugation using Heinz numbers.
%Y A351980 A195017 = # of even parts - # of odd parts.
%Y A351980 A257991 counts odd parts, conjugate A344616.
%Y A351980 A257992 counts even parts, conjugate A350847.
%Y A351980 A316524 = alternating sum of prime indices.
%Y A351980 A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A351980 A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A351980 Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350841, A350849, A350941, A350942, A350950, A350951.
%K A351980 nonn
%O A351980 1,2
%A A351980 _Gus Wiseman_, Mar 14 2022