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

1, 15, 33, 35, 51, 55, 66, 69, 70, 77, 85, 91, 93, 95, 99, 102, 105, 110, 119, 123, 132, 138, 140, 141, 143, 145, 153, 154, 155, 161, 165, 170, 175, 177, 182, 186, 187, 190, 201, 203, 204, 205, 207, 209, 210, 215, 217, 219, 220, 221, 231, 238, 245, 246, 247, 249

Offset: 1

Author

Gus Wiseman, Sep 25 2019

Keywords

• nonn

Comments

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:
    1: {}
   15: {2,3}
   33: {2,5}
   35: {3,4}
   51: {2,7}
   55: {3,5}
   66: {1,2,5}
   69: {2,9}
   70: {1,3,4}
   77: {4,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
   95: {3,8}
   99: {2,2,5}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  119: {4,7}
  123: {2,13}
  132: {1,1,2,5}
		

Crossrefs

The enumeration of these partitions by sum is A327779.

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 a multiple their sum are A327783.

Cf. A056239, A074761, A112798, A290103, A316413, A326841, A327781.

Programs

Maple

q:= n-> (l-> is(ilcm(l[])>add(j, j=l)))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..250])[];  # Alois P. Heinz, Sep 27 2019

Mathematica

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

Formula

A290103(a(k)) > A056239(a(k)).