A352874 - OEIS

%I #7 May 15 2022 11:50:37

%S 3,5,7,9,11,13,15,17,18,19,21,23,25,27,29,30,31,33,35,37,39,41,42,43,

%T 45,47,49,50,51,53,54,55,57,59,61,63,65,66,67,69,70,71,73,75,77,78,79,

%U 81,83,85,87,89,90,91,93,95,97,98,99,101,102,103,105,107,109

%N Heinz numbers of integer partitions with positive crank, counted by A001522.

%C 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.

%C 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.]

%F Complement of A342192 in A352873.

%e The terms together with their prime indices begin:

%e       3: (2)         30: (3,2,1)     54: (2,2,2,1)

%e       5: (3)         31: (11)        55: (5,3)

%e       7: (4)         33: (5,2)       57: (8,2)

%e       9: (2,2)       35: (4,3)       59: (17)

%e      11: (5)         37: (12)        61: (18)

%e      13: (6)         39: (6,2)       63: (4,2,2)

%e      15: (3,2)       41: (13)        65: (6,3)

%e      17: (7)         42: (4,2,1)     66: (5,2,1)

%e      18: (2,2,1)     43: (14)        67: (19)

%e      19: (8)         45: (3,2,2)     69: (9,2)

%e      21: (4,2)       47: (15)        70: (4,3,1)

%e      23: (9)         49: (4,4)       71: (20)

%e      25: (3,3)       50: (3,3,1)     73: (21)

%e      27: (2,2,2)     51: (7,2)       75: (3,3,2)

%e      29: (10)        53: (16)        77: (5,4)

%t ck[y_]:=With[{w=Count[y,1]},If[w==0,Max@@y,Count[y,_?(#>w&)]-w]];

%t Select[Range[100],ck[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]>0&]

%Y * = unproved

%Y These partitions are counted by A001522.

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

%Y The case of nonnegative crank is A352873, counted by A064428.

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A001222 counts prime indices, distinct A001221.

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

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

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

%Y A115720 and A115994 count partitions by their Durfee square.

%Y A122111 represents partition conjugation using Heinz numbers.

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 09 2022