A327783 - OEIS

%I #4 Oct 16 2019 13:09:53

%S 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 89,97,101,103,107,109,113,127,131,137,139,149,151,154,157,163,165,

%U 167,173,179,181,190,191,193,197,198,199,211,223,227,229,233,239,241

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

%C First differs from A319333 in having 154.

%C First nonsquarefree term is 198.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%F A056239(a(k)) | A290103(a(k)).

%e The sequence of terms together with their prime indices begins:

%e     2: {1}

%e     3: {2}

%e     5: {3}

%e     7: {4}

%e    11: {5}

%e    13: {6}

%e    17: {7}

%e    19: {8}

%e    23: {9}

%e    29: {10}

%e    30: {1,2,3}

%e    31: {11}

%e    37: {12}

%e    41: {13}

%e    43: {14}

%e    47: {15}

%e    53: {16}

%e    59: {17}

%e    61: {18}

%e    67: {19}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[2,100],Divisible[LCM@@primeMS[#],Total[primeMS[#]]]&]

%Y The enumeration of these partitions by sum is A327778.

%Y Heinz numbers of partitions whose LCM is twice their sum are A327775.

%Y Heinz numbers of partitions whose LCM is less than their sum are A327776.

%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