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%I #10 May 15 2022 11:50:12
%S 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,
%T 35,37,38,39,41,42,43,45,46,47,49,50,51,53,54,55,57,58,59,61,62,63,65,
%U 66,67,69,70,71,73,74,75,77,78,79,81,82,83,85,86,87,89,90
%N Heinz numbers of integer partitions with nonnegative crank, counted by A064428.
%C First differs from A042968, A059557, and A195291 in lacking 2 and having 100.
%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 Union of A352874 and A342192.
%e The terms together with their prime indices begin:
%e 1: () 22: (5,1) 42: (4,2,1)
%e 3: (2) 23: (9) 43: (14)
%e 5: (3) 25: (3,3) 45: (3,2,2)
%e 6: (2,1) 26: (6,1) 46: (9,1)
%e 7: (4) 27: (2,2,2) 47: (15)
%e 9: (2,2) 29: (10) 49: (4,4)
%e 10: (3,1) 30: (3,2,1) 50: (3,3,1)
%e 11: (5) 31: (11) 51: (7,2)
%e 13: (6) 33: (5,2) 53: (16)
%e 14: (4,1) 34: (7,1) 54: (2,2,2,1)
%e 15: (3,2) 35: (4,3) 55: (5,3)
%e 17: (7) 37: (12) 57: (8,2)
%e 18: (2,2,1) 38: (8,1) 58: (10,1)
%e 19: (8) 39: (6,2) 59: (17)
%e 21: (4,2) 41: (13) 61: (18)
%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 A064428.
%Y The case of zero crank is A342192, counted by A064410.
%Y The case of positive crank is A352874.
%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 A238394 counts reversed partitions without a fixed point, ranked by A352830.
%Y Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831.
%K nonn
%O 1,2
%A _Gus Wiseman_, Apr 09 2022