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

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 68, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 100, 104, 106, 108, 111, 112

Offset

1,1

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..64.

Example

The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}

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..120])[];  # 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 A327781.

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

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, A316413, A326841, A327779.

Sequence in context: A110615 A060679 A051234 * A102554 A371862 A070810

Adjacent sequences: A327773 A327774 A327775 * A327777 A327778 A327779

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved