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A366532 Heinz numbers of integer partitions with at least one even and odd part.

%I A366532 #5 Oct 16 2023 23:26:50
%S A366532 6,12,14,15,18,24,26,28,30,33,35,36,38,42,45,48,51,52,54,56,58,60,65,
%T A366532 66,69,70,72,74,75,76,77,78,84,86,90,93,95,96,98,99,102,104,105,106,
%U A366532 108,112,114,116,119,120,122,123,126,130,132,135,138,140,141,142
%N A366532 Heinz numbers of integer partitions with at least one even and odd part.
%C A366532 These partitions are counted by A006477.
%C A366532 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 A366532 Intersection of A324929 and A366322.
%e A366532 The terms together with their prime indices begin:
%e A366532     6: {1,2}
%e A366532    12: {1,1,2}
%e A366532    14: {1,4}
%e A366532    15: {2,3}
%e A366532    18: {1,2,2}
%e A366532    24: {1,1,1,2}
%e A366532    26: {1,6}
%e A366532    28: {1,1,4}
%e A366532    30: {1,2,3}
%e A366532    33: {2,5}
%e A366532    35: {3,4}
%e A366532    36: {1,1,2,2}
%e A366532    38: {1,8}
%e A366532    42: {1,2,4}
%e A366532    45: {2,2,3}
%e A366532    48: {1,1,1,1,2}
%t A366532 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366532 Select[Range[100],Or@@EvenQ/@prix[#]&&Or@@OddQ/@prix[#]&]
%Y A366532 These partitions are counted by A006477.
%Y A366532 Just even: A324929, counted by A047967.
%Y A366532 Just odd: A366322, counted by A086543 (even bisection of A182616).
%Y A366532 A031368 lists primes of odd index, even A031215.
%Y A366532 A066207 ranks partitions with all even parts, counted by A035363.
%Y A366532 A066208 ranks partitions with all odd parts, counted by A000009.
%Y A366532 A112798 lists prime indices, sum A056239.
%Y A366532 A257991 counts odd prime indices, distinct A324966.
%Y A366532 A257992 counts even prime indices, distinct A324967.
%Y A366532 Cf. A000720, A001222, A003963, A026804, A244991, A257992.
%K A366532 nonn
%O A366532 1,1
%A A366532 _Gus Wiseman_, Oct 16 2023