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 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 Table of n, a(n) for n=1..65. 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. Cf. A065770, A093641, A118199, A188674, A252464, A257990, A325163, A325169, A344609, A352828, A352831. Sequence in context: A160931 A160924 A063280 * A249123 A094042 A248196 Adjacent sequences: A352871 A352872 A352873 * A352875 A352876 A352877 KEYWORD nonn AUTHOR Gus Wiseman, Apr 09 2022 STATUS approved

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