A327776 - OEIS

%I #6 Sep 27 2019 14:59:13

%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