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

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Author

Gus Wiseman, Sep 25 2019

Keywords

• nonn

Comments

First differs from A319333 in having 154.

First nonsquarefree term is 198.

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

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   53: {16}
   59: {17}
   61: {18}
   67: {19}
		

Crossrefs

The enumeration of these partitions by sum is A327778.

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

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

Heinz numbers of partitions whose LCM is greater than their sum are A327784.

Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327779.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[LCM@@primeMS[#],Total[primeMS[#]]]&]

Formula

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