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

154, 190, 435, 580, 714, 952, 1118, 1287, 1430, 1653, 1716, 1815, 1935, 2067, 2150, 2204, 2254, 2288, 2415, 2475, 2580, 2756, 2898, 2970, 3220, 3300, 3440, 3710, 3864, 3960, 3975, 4770, 5152, 5280, 5300, 6360, 6461, 6897, 7514, 8307, 8480, 8619, 8695, 8778

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:
   154: {1,4,5}
   190: {1,3,8}
   435: {2,3,10}
   580: {1,1,3,10}
   714: {1,2,4,7}
   952: {1,1,1,4,7}
  1118: {1,6,14}
  1287: {2,2,5,6}
  1430: {1,3,5,6}
  1653: {2,8,10}
  1716: {1,1,2,5,6}
  1815: {2,3,5,5}
  1935: {2,2,3,14}
  2067: {2,6,16}
  2150: {1,3,3,14}
  2204: {1,1,8,10}
  2254: {1,4,4,9}
  2288: {1,1,1,1,5,6}
  2415: {2,3,4,9}
  2475: {2,2,3,3,5}
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

The enumeration of these partitions by sum is A327780.

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.

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

Cf. A056239, A074761, A112798, A290103, A326841 , A327778.

Programs

Maple

q:= n-> (l-> is(ilcm(l[])=2*add(j, j=l)))(map(i->
        numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..10000])[];  # Alois P. Heinz, Sep 27 2019

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],LCM@@primeMS[#]==2*Total[primeMS[#]]&]

Formula

A290103(a(k)) = 2 * A056239(a(k)).