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A366529 Heinz numbers of integer partitions of even numbers with at least one even part.

%I A366529 #5 Oct 16 2023 13:42:43
%S A366529 3,7,9,12,13,19,21,27,28,29,30,36,37,39,43,48,49,52,53,57,61,63,66,70,
%T A366529 71,75,76,79,81,84,87,89,90,91,101,102,107,108,111,112,113,116,117,
%U A366529 120,129,130,131,133,138,139,144,147,148,151,154,156,159,163,165
%N A366529 Heinz numbers of integer partitions of even numbers with at least one even part.
%C A366529 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 A366529 The terms together with their prime indices begin:
%e A366529    3: {2}
%e A366529    7: {4}
%e A366529    9: {2,2}
%e A366529   12: {1,1,2}
%e A366529   13: {6}
%e A366529   19: {8}
%e A366529   21: {2,4}
%e A366529   27: {2,2,2}
%e A366529   28: {1,1,4}
%e A366529   29: {10}
%e A366529   30: {1,2,3}
%e A366529   36: {1,1,2,2}
%e A366529   37: {12}
%e A366529   39: {2,6}
%e A366529   43: {14}
%e A366529   48: {1,1,1,1,2}
%t A366529 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366529 Select[Range[100],EvenQ[Total[prix[#]]]&&Or@@EvenQ/@prix[#]&]
%Y A366529 The complement is counted by A047967.
%Y A366529 For all even parts we have A066207, counted by A035363, odd A066208.
%Y A366529 Not requiring an even part gives A300061.
%Y A366529 For odd instead of even we have A300063.
%Y A366529 Not requiring even sum gives A324929.
%Y A366529 Partitions of this type are counted by A366527.
%Y A366529 A112798 list prime indices, sum A056239.
%Y A366529 A257991 counts odd prime indices, distinct A324966.
%Y A366529 A257992 counts even prime indices, distinct A324967.
%Y A366529 Cf. A000720, A001222, A003963, A033844, A086543, A324927, A358137, A366530.
%K A366529 nonn
%O A366529 1,1
%A A366529 _Gus Wiseman_, Oct 16 2023