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A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

%I A365003 #8 Aug 24 2023 10:03:05
%S A365003 1,4,9,25,36,48,49,100,121,160,169,196,225,289,361,441,448,484,529,
%T A365003 567,676,750,810,841,900,961,1080,1089,1156,1200,1225,1369,1408,1440,
%U A365003 1444,1521,1681,1764,1849,1920,2116,2209,2268,2352,2601,2809,3024,3025,3159
%N A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.
%C A365003 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 A365003 A056239(a(n)) = 2*A066328(a(n)).
%e A365003 The prime indices of 750 are {1,2,3,3,3}, with sum 12, while the distinct prime indices {1,2,3} have sum 6, so 750 is in the sequence.
%e A365003 The terms together with their prime indices begin:
%e A365003      1: {}
%e A365003      4: {1,1}
%e A365003      9: {2,2}
%e A365003     25: {3,3}
%e A365003     36: {1,1,2,2}
%e A365003     48: {1,1,1,1,2}
%e A365003     49: {4,4}
%e A365003    100: {1,1,3,3}
%e A365003    121: {5,5}
%e A365003    160: {1,1,1,1,1,3}
%e A365003    169: {6,6}
%e A365003    196: {1,1,4,4}
%e A365003    225: {2,2,3,3}
%e A365003    289: {7,7}
%e A365003    361: {8,8}
%e A365003    441: {2,2,4,4}
%e A365003    448: {1,1,1,1,1,1,4}
%t A365003 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A365003 Select[Range[1000],Total[prix[#]]==2*Total[Union[prix[#]]]&]
%Y A365003 The LHS is A056239 (sum of prime indices).
%Y A365003 The RHS is twice A066328.
%Y A365003 Partitions of this type are counted by A364910.
%Y A365003 A000041 counts integer partitions, strict A000009.
%Y A365003 A001222 counts prime indices, distinct A001221.
%Y A365003 A112798 lists prime indices, distinct A304038.
%Y A365003 A116861 counts partitions by sum and sum of distinct parts.
%Y A365003 A323092 counts double-free partitions, ranks A320340.
%Y A365003 Cf. A364350, A364839, A364906, A364907, A364911, A364916.
%K A365003 nonn
%O A365003 1,2
%A A365003 _Gus Wiseman_, Aug 23 2023