A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset

1,1

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

Links

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

Example

The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], MemberQ[Divide@@@Partition[primeptn[#], 2, 1], 2]&]

Crossrefs

The complement is A350838, counted by A350837.

The strict complement is counted by A350840.

These partitions are counted by A350846.

A000041 = integer partitions.

A000045 = sets containing n with all differences > 2.

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

A116931 = partitions with no successions, ranked by A319630.

A116932 = partitions with differences != 1 or 2, strict A025157.

A323092 = double-free integer partitions, ranked by A320340.

A325160 ranks strict partitions with no successions, counted by A003114.

A350839 = partitions with gaps and conjugate gaps, ranked by A350841.

Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

Sequence in context: A046411 A364348 A364537 * A315723 A361908 A160594

Adjacent sequences: A350842 A350843 A350844 * A350846 A350847 A350848

Keyword

nonn

Author

Gus Wiseman, Jan 20 2022

Status

approved