%I
%S 4,6,8,9,10,12,14,16,18,20,21,22,24,25,26,27,28,32,34,36,38,39,40,42,
%T 44,45,46,48,49,50,52,54,56,57,58,60,62,63,64,65,68,72,74,75,76,78,80,
%U 81,82,84,86,87,88,90,92,94,96,98,100,104,106,108,111,112
%N Heinz numbers of integer partitions whose LCM is less than their sum.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e The sequence of terms together with their prime indices begins:
%e 4: {1,1}
%e 6: {1,2}
%e 8: {1,1,1}
%e 9: {2,2}
%e 10: {1,3}
%e 12: {1,1,2}
%e 14: {1,4}
%e 16: {1,1,1,1}
%e 18: {1,2,2}
%e 20: {1,1,3}
%e 21: {2,4}
%e 22: {1,5}
%e 24: {1,1,1,2}
%e 25: {3,3}
%e 26: {1,6}
%e 27: {2,2,2}
%e 28: {1,1,4}
%e 32: {1,1,1,1,1}
%e 34: {1,7}
%e 36: {1,1,2,2}
%p q:= n> (l> is(ilcm(l[])<add(j, j=l)))(map(i>
%p numtheory[pi](i[1])$i[2], ifactors(n)[2])):
%p select(q, [$1..120])[]; # _Alois P. Heinz_, Sep 27 2019
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[2,100],LCM@@primeMS[#]<Total[primeMS[#]]&]
%Y The enumeration of these partitions by sum is A327781.
%Y Heinz numbers of partitions whose LCM is twice their sum are A327775.
%Y Heinz numbers of partitions whose LCM is a multiple their sum are A327783.
%Y Heinz numbers of partitions whose LCM is greater than their sum are A327784.
%Y Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327779.
%K nonn
%O 1,1
%A _Gus Wiseman_, Sep 25 2019
