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