A352827 Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.

2, 4, 8, 9, 15, 16, 18, 21, 27, 30, 32, 33, 36, 39, 42, 45, 51, 54, 57, 60, 63, 64, 66, 69, 72, 78, 81, 84, 87, 90, 93, 99, 102, 108, 111, 114, 117, 120, 123, 125, 126, 128, 129, 132, 135, 138, 141, 144, 153, 156, 159, 162, 168, 171, 174, 175, 177, 180, 183

Offset: 1

Author

Gus Wiseman, Apr 06 2022

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
    2: (1)
    4: (1,1)
    8: (1,1,1)
    9: (2,2)
   15: (3,2)
   16: (1,1,1,1)
   18: (2,2,1)
   21: (4,2)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   33: (5,2)
   36: (2,2,1,1)
   39: (6,2)
   42: (4,2,1)
   45: (3,2,2)
   51: (7,2)
   54: (2,2,2,1)
For example, the partition (3,2,2) with Heinz number 45 has a fixed point at position 2, so 45 is in the sequence.
		

Crossrefs

* = unproved

*These partitions are counted by A001522, strict A352829.

*The complement is A352826, counted by A064428.

The complement reverse version is A352830, counted by A238394.

The reverse version is A352872, counted by A238395

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

A008290 counts permutations by fixed points, unfixed A098825.

A056239 adds up prime indices, row sums of A112798 and A296150.

A115720 and A115994 count partitions by their Durfee square.

A122111 represents partition conjugation using Heinz numbers.

A124010 gives prime signature, sorted A118914, conjugate rank A238745.

A238349 counts compositions by fixed points, complement A352523.

A238352 counts reversed partitions by fixed points, rank statistic A352822.

A352828 counts strict partitions without a fixed point.

A352833 counts partitions by fixed points.

Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824 (characteristic function), A352825, A352831.

Programs

Mathematica

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==1&]