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A327783 Heinz numbers of integer partitions whose LCM is a multiple of their sum.

%I A327783 #4 Oct 16 2019 13:09:53
%S A327783 2,3,5,7,11,13,17,19,23,29,30,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A327783 89,97,101,103,107,109,113,127,131,137,139,149,151,154,157,163,165,
%U A327783 167,173,179,181,190,191,193,197,198,199,211,223,227,229,233,239,241
%N A327783 Heinz numbers of integer partitions whose LCM is a multiple of their sum.
%C A327783 First differs from A319333 in having 154.
%C A327783 First nonsquarefree term is 198.
%C A327783 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A327783 A056239(a(k)) | A290103(a(k)).
%e A327783 The sequence of terms together with their prime indices begins:
%e A327783     2: {1}
%e A327783     3: {2}
%e A327783     5: {3}
%e A327783     7: {4}
%e A327783    11: {5}
%e A327783    13: {6}
%e A327783    17: {7}
%e A327783    19: {8}
%e A327783    23: {9}
%e A327783    29: {10}
%e A327783    30: {1,2,3}
%e A327783    31: {11}
%e A327783    37: {12}
%e A327783    41: {13}
%e A327783    43: {14}
%e A327783    47: {15}
%e A327783    53: {16}
%e A327783    59: {17}
%e A327783    61: {18}
%e A327783    67: {19}
%t A327783 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A327783 Select[Range[2,100],Divisible[LCM@@primeMS[#],Total[primeMS[#]]]&]
%Y A327783 The enumeration of these partitions by sum is A327778.
%Y A327783 Heinz numbers of partitions whose LCM is twice their sum are A327775.
%Y A327783 Heinz numbers of partitions whose LCM is less than their sum are A327776.
%Y A327783 Heinz numbers of partitions whose LCM is greater than their sum are A327784.
%Y A327783 Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327779.
%K A327783 nonn
%O A327783 1,1
%A A327783 _Gus Wiseman_, Sep 25 2019