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A327776 Heinz numbers of integer partitions whose LCM is less than their sum. 4

%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

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Last modified July 25 18:32 EDT 2021. Contains 346291 sequences. (Running on oeis4.)