A356736 Heinz numbers of integer partitions with no neighborless parts.

1, 6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 462, 480, 486, 525, 539, 540

Offset

1,2

Comments

First differs from A066312 in having 1 and lacking 462.

First differs from A104210 in having 1 and lacking 42.

A part x is neighborless iff neither x - 1 nor x + 1 are parts.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Links

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

Example

The terms together with their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  30: {1,2,3}
  35: {3,4}
  36: {1,1,2,2}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  75: {2,3,3}
  77: {4,5}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Function[ptn, !Or@@Table[!MemberQ[ptn, x-1]&&!MemberQ[ptn, x+1], {x, Union[ptn]}]]@*primeMS]

Crossrefs

These partitions are counted by A355394.

The singleton case is the complement of A356237.

The singleton case is counted by A355393, complement A356235.

The strict complement is A356606, counted by A356607.

The complement is A356734, counted by A356236.

A000041 counts integer partitions, strict A000009.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together the prime indices of n.

A007690 counts partitions with no singletons, complement A183558.

A056239 adds up prime indices, row sums of A112798, lengths A001222.

A073491 lists numbers with gapless prime indices, complement A073492.

Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.

Sequence in context: A091011 A324771 A104210 * A066312 A309944 A212308

Adjacent sequences: A356733 A356734 A356735 * A356737 A356738 A356739

Keyword

nonn

Author

Gus Wiseman, Aug 31 2022

Status

approved