

A352873


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


9



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
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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.
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



FORMULA



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 positive crank is A352874.
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.


KEYWORD

nonn


AUTHOR



STATUS

approved



