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

1,1

Comments

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

Links

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

Formula

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

Example

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}

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[#]]&]

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.

Sequence in context: A049515 A049519 A214475 * A053243 A320708 A261377

Adjacent sequences: A327772 A327773 A327774 * A327776 A327777 A327778

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved