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A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.

%I A326838 #6 Jul 29 2019 13:54:42
%S A326838 30,84,264,273,286,325,351,364,390,441,490,525,624,756,784,810,840,
%T A326838 874,900,988,1000,1173,1197,1254,1330,1425,1495,1632,1771,2079,2156,
%U A326838 2178,2204,2294,2310,2420,2475,2750,2958,3219,3393,3648,3726,3770,3864,3944,4042
%N A326838 Heinz numbers of non-constant integer partitions whose length and maximum both divide their sum.
%C A326838 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326838 The enumeration of these partitions by sum is given by A326852.
%e A326838 The sequence of terms together with their prime indices begins:
%e A326838     30: {1,2,3}
%e A326838     84: {1,1,2,4}
%e A326838    264: {1,1,1,2,5}
%e A326838    273: {2,4,6}
%e A326838    286: {1,5,6}
%e A326838    325: {3,3,6}
%e A326838    351: {2,2,2,6}
%e A326838    364: {1,1,4,6}
%e A326838    390: {1,2,3,6}
%e A326838    441: {2,2,4,4}
%e A326838    490: {1,3,4,4}
%e A326838    525: {2,3,3,4}
%e A326838    624: {1,1,1,1,2,6}
%e A326838    756: {1,1,2,2,2,4}
%e A326838    784: {1,1,1,1,4,4}
%e A326838    810: {1,2,2,2,2,3}
%e A326838    840: {1,1,1,2,3,4}
%e A326838    874: {1,8,9}
%e A326838    900: {1,1,2,2,3,3}
%e A326838    988: {1,1,6,8}
%t A326838 Select[Range[1000],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},!SameQ@@y&&Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]
%Y A326838 The possibly constant case is A326837.
%Y A326838 Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848, A326851.
%K A326838 nonn
%O A326838 1,1
%A A326838 _Gus Wiseman_, Jul 26 2019