A352873 Heinz numbers of integer partitions with nonnegative crank, counted by A064428.

1, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90

Offset

1,2

Comments

First differs from A042968, A059557, and A195291 in lacking 2 and having 100.

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.

The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's). [Definition copied from A342192; see A064428 for a different wording.]

Links

Table of n, a(n) for n=1..67.

Formula

Union of A352874 and A342192.

Example

The terms together with their prime indices begin:
     1: ()         22: (5,1)      42: (4,2,1)
     3: (2)        23: (9)        43: (14)
     5: (3)        25: (3,3)      45: (3,2,2)
     6: (2,1)      26: (6,1)      46: (9,1)
     7: (4)        27: (2,2,2)    47: (15)
     9: (2,2)      29: (10)       49: (4,4)
    10: (3,1)      30: (3,2,1)    50: (3,3,1)
    11: (5)        31: (11)       51: (7,2)
    13: (6)        33: (5,2)      53: (16)
    14: (4,1)      34: (7,1)      54: (2,2,2,1)
    15: (3,2)      35: (4,3)      55: (5,3)
    17: (7)        37: (12)       57: (8,2)
    18: (2,2,1)    38: (8,1)      58: (10,1)
    19: (8)        39: (6,2)      59: (17)
    21: (4,2)      41: (13)       61: (18)

Mathematica

ck[y_]:=With[{w=Count[y, 1]}, If[w==0, Max@@y, Count[y, _?(#>w&)]-w]];
Select[Range[100], ck[Reverse[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]]>=0&]

Crossrefs

* = unproved

These partitions are counted by A064428.

The case of zero crank is A342192, counted by A064410.

The case of positive crank is A352874.

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

*A001522 counts partitions with a fixed point, ranked by A352827.

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

*A064428 counts partitions without a fixed point, ranked by A352826.

A115720 and A115994 count partitions by their Durfee square.

A122111 represents partition conjugation using Heinz numbers.

A238394 counts reversed partitions without a fixed point, ranked by A352830.

Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.

Sequence in context: A186145 A364058 A335740 * A047984 A288513 A125236

Adjacent sequences: A352870 A352871 A352872 * A352874 A352875 A352876

Keyword

nonn

Author

Gus Wiseman, Apr 09 2022

Status

approved