A322109 Heinz numbers of integer partitions that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons.

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 189, 192, 196, 198, 200, 210

Offset

1,2

Comments

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

Also Heinz numbers of partitions whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

Links

Table of n, a(n) for n=1..59.

Formula

A061395(a(n)) <= A056239(a(n))/2.

Example

Each term paired with its Heinz partition and a realizing set multipartition with no singletons:
   1:      (): {}
   4:    (11): {{1,2}}
   8:   (111): {{1,2,3}}
   9:    (22): {{1,2},{1,2}}
  12:   (211): {{1,2},{1,3}}
  16:  (1111): {{1,2,3,4}}
  18:   (221): {{1,2},{1,2,3}}
  24:  (2111): {{1,2},{1,3,4}}
  25:    (33): {{1,2},{1,2},{1,2}}
  27:   (222): {{1,2,3},{1,2,3}}
  30:   (321): {{1,2},{1,2},{1,3}}
  32: (11111): {{1,2,3,4,5}}
  36:  (2211): {{1,2},{1,2,3,4}}
  40:  (3111): {{1,2},{1,3},{1,4}}

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
sqnopfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqnopfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], !PrimeQ[#]&&SquareFreeQ[#]&]}]]
Select[Range[100], Length[sqnopfacs[Times@@Prime/@nrmptn[#]]]>0&]

Crossrefs

These partitions are counted by A110618.

The even-weight version is A320924.

The conjugate case of equality is A340387.

The conjugate version is A344291.

The opposite conjugate version is A344296.

The opposite version is A344414.

The case of equality is A344415.

The opposite even-weight version is A344416.

A000070 counts non-multigraphical partitions.

A025065 counts palindromic partitions.

A035363 counts partitions into even parts.

A056239 adds up prime indices, row sums of A112798.

A058696 counts partitions of even numbers, ranked by A300061.

A334201 adds up all prime indices except the greatest.

Cf. A000041, A000569, A265640, A283877, A306005, A318361, A320922, A320923, A320925.

Sequence in context: A053443 A360070 A048098 * A122145 A328014 A359869

Adjacent sequences: A322106 A322107 A322108 * A322110 A322111 A322112

Keyword

nonn

Author

Gus Wiseman, Nov 26 2018

Status

approved