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

1,2

Comments

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

Links

Table of n, a(n) for n=1..56.

Formula

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

Example

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}

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[#]]&]

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.

Sequence in context: A154369 A243592 A089966 * A339562 A338468 A337984

Adjacent sequences: A327781 A327782 A327783 * A327785 A327786 A327787

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved