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%I #11 Feb 22 2022 23:42:09
%S 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,37,
%T 40,41,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,67,68,71,72,73,75,
%U 76,79,80,81,83,88,89,92,96,97,98,99,101,103,104,107,108,109
%N Heinz numbers of Look-and-Say partitions. Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.
%C First differs from A130091 (Wilf partitions) in having 216.
%C See A239455 for the definition of Look-and-Say partitions.
%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.
%e The terms together with their prime indices begin:
%e 1: () 20: (3,1,1) 47: (15)
%e 2: (1) 23: (9) 48: (2,1,1,1,1)
%e 3: (2) 24: (2,1,1,1) 49: (4,4)
%e 4: (1,1) 25: (3,3) 50: (3,3,1)
%e 5: (3) 27: (2,2,2) 52: (6,1,1)
%e 7: (4) 28: (4,1,1) 53: (16)
%e 8: (1,1,1) 29: (10) 54: (2,2,2,1)
%e 9: (2,2) 31: (11) 56: (4,1,1,1)
%e 11: (5) 32: (1,1,1,1,1) 59: (17)
%e 12: (2,1,1) 37: (12) 61: (18)
%e 13: (6) 40: (3,1,1,1) 63: (4,2,2)
%e 16: (1,1,1,1) 41: (13) 64: (1,1,1,1,1,1)
%e 17: (7) 43: (14) 67: (19)
%e 18: (2,2,1) 44: (5,1,1) 68: (7,1,1)
%e 19: (8) 45: (3,2,2) 71: (20)
%e For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).
%t Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]!={}&]
%Y The Wilf case (distinct multiplicities) is A130091, counted by A098859.
%Y The complement of the Wilf case is A130092, counted by A336866.
%Y These partitions are counted by A239455.
%Y A variant for runs is A351201, counted by A351203 (complement A351204).
%Y The complement is A351295, counted by A351293.
%Y A032020 = number of binary expansions with distinct run-lengths.
%Y A044813 = numbers whose binary expansion has all distinct run-lengths.
%Y A056239 = sum of prime indices, row sums of A112798.
%Y A165413 = number of run-lengths in binary expansion, for all runs A297770.
%Y A181819 = Heinz number of prime signature (prime shadow).
%Y A182850/A323014 = frequency depth, counted by A225485/A325280.
%Y A320922 ranks graphical partitions, complement A339618, counted by A000569.
%Y A329739 = compositions with all distinct run-lengths, for all runs A351013.
%Y A333489 ranks anti-runs, complement A348612.
%Y A351017 = binary words with all distinct run-lengths, for all runs A351016.
%Y A351292 = patterns with all distinct run-lengths, for all runs A351200.
%Y Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 15 2022
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