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A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

%I A366530 #7 Oct 16 2023 20:12:41
%S A366530 4,10,12,16,22,25,28,30,34,36,40,46,48,52,55,62,64,66,70,75,76,82,84,
%T A366530 85,88,90,94,100,102,108,112,115,116,118,120,121,130,134,136,138,144,
%U A366530 146,148,154,155,156,160,165,166,172,175,184,186,187,190,192,194,196
%N A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.
%C A366530 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.
%e A366530 The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
%e A366530     4: {1,1}
%e A366530    10: {1,3}
%e A366530    12: {1,1,2}
%e A366530    16: {1,1,1,1}
%e A366530    22: {1,5}
%e A366530    25: {3,3}
%e A366530    28: {1,1,4}
%e A366530    30: {1,2,3}
%e A366530    34: {1,7}
%e A366530    36: {1,1,2,2}
%e A366530    40: {1,1,1,3}
%e A366530    46: {1,9}
%e A366530    48: {1,1,1,1,2}
%e A366530    52: {1,1,6}
%e A366530    55: {3,5}
%e A366530    62: {1,11}
%e A366530    64: {1,1,1,1,1,1}
%t A366530 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366530 Select[Range[100], EvenQ[Total[prix[#]]]&&Or@@OddQ/@prix[#]&]
%Y A366530 These partitions are counted by A182616, even bisection of A086543.
%Y A366530 Not requiring at least one odd part gives A300061.
%Y A366530 Allowing partitions of odd numbers gives A366322.
%Y A366530 A031368 lists primes of odd index.
%Y A366530 A066207 ranks partitions with all even parts, counted by A035363.
%Y A366530 A066208 ranks partitions with all odd parts, counted by A000009.
%Y A366530 A112798 list prime indices, sum A056239.
%Y A366530 A257991 counts odd prime indices, distinct A324966.
%Y A366530 Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.
%K A366530 nonn
%O A366530 1,1
%A A366530 _Gus Wiseman_, Oct 16 2023