login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A352874 Heinz numbers of integer partitions with positive crank, counted by A001522. 9
3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101, 102, 103, 105, 107, 109 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
Complement of A342192 in A352873.
EXAMPLE
The terms together with their prime indices begin:
3: (2) 30: (3,2,1) 54: (2,2,2,1)
5: (3) 31: (11) 55: (5,3)
7: (4) 33: (5,2) 57: (8,2)
9: (2,2) 35: (4,3) 59: (17)
11: (5) 37: (12) 61: (18)
13: (6) 39: (6,2) 63: (4,2,2)
15: (3,2) 41: (13) 65: (6,3)
17: (7) 42: (4,2,1) 66: (5,2,1)
18: (2,2,1) 43: (14) 67: (19)
19: (8) 45: (3,2,2) 69: (9,2)
21: (4,2) 47: (15) 70: (4,3,1)
23: (9) 49: (4,4) 71: (20)
25: (3,3) 50: (3,3,1) 73: (21)
27: (2,2,2) 51: (7,2) 75: (3,3,2)
29: (10) 53: (16) 77: (5,4)
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 A001522.
The case of zero crank is A342192, counted by A064410.
The case of nonnegative crank is A352873, counted by A064428.
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.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Sequence in context: A160931 A160924 A063280 * A249123 A094042 A248196
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 09 2022
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 12 16:59 EDT 2024. Contains 371635 sequences. (Running on oeis4.)