A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.

1, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 97

Offset

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

The terms together with their prime indices begin:
      1: ()          24: (2,1,1,1)     47: (15)
      3: (2)         25: (3,3)         48: (2,1,1,1,1)
      5: (3)         26: (6,1)         49: (4,4)
      6: (2,1)       28: (4,1,1)       50: (3,3,1)
      7: (4)         29: (10)          52: (6,1,1)
     10: (3,1)       31: (11)          53: (16)
     11: (5)         34: (7,1)         55: (5,3)
     12: (2,1,1)     35: (4,3)         56: (4,1,1,1)
     13: (6)         37: (12)          58: (10,1)
     14: (4,1)       38: (8,1)         59: (17)
     17: (7)         40: (3,1,1,1)     61: (18)
     19: (8)         41: (13)          62: (11,1)
     20: (3,1,1)     43: (14)          65: (6,3)
     22: (5,1)       44: (5,1,1)       67: (19)
     23: (9)         46: (9,1)         68: (7,1,1)

Mathematica

pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Select[Range[100], pq[Reverse[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]]==0&]

Crossrefs

* = unproved

*These partitions are counted by A064428, strict A352828.

The complement is A352827.

The reverse version is A352830, counted by A238394.

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

*A001522 counts partitions with a fixed point.

A008290 counts permutations by fixed points, nonfixed 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.

A238349 counts compositions by fixed points, complement A352523.

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

A238395 counts reversed partitions with a fixed point, ranked by A352872.

A352833 counts partitions by fixed points.

Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831, A352832.

Sequence in context: A028983 A232682 A364289 * A335657 A038550 A204232

Adjacent sequences: A352823 A352824 A352825 * A352827 A352828 A352829

Keyword

nonn

Author

Gus Wiseman, Apr 06 2022

Status

approved