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A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

%I A364058 #7 Jul 15 2023 05:42:35
%S A364058 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 A364058 35,36,37,38,39,41,42,43,45,46,47,49,50,51,53,54,55,57,58,59,60,61,62,
%U A364058 63,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83,84,85,86
%N A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.
%C A364058 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 A364058 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%F A364058 A360005(a(n)) > 1.
%F A364058 A360459(a(n)) > 2.
%e A364058 The terms together with their prime indices begin:
%e A364058      3: {2}        23: {9}          42: {1,2,4}
%e A364058      5: {3}        25: {3,3}        43: {14}
%e A364058      6: {1,2}      26: {1,6}        45: {2,2,3}
%e A364058      7: {4}        27: {2,2,2}      46: {1,9}
%e A364058      9: {2,2}      29: {10}         47: {15}
%e A364058     10: {1,3}      30: {1,2,3}      49: {4,4}
%e A364058     11: {5}        31: {11}         50: {1,3,3}
%e A364058     13: {6}        33: {2,5}        51: {2,7}
%e A364058     14: {1,4}      34: {1,7}        53: {16}
%e A364058     15: {2,3}      35: {3,4}        54: {1,2,2,2}
%e A364058     17: {7}        36: {1,1,2,2}    55: {3,5}
%e A364058     18: {1,2,2}    37: {12}         57: {2,8}
%e A364058     19: {8}        38: {1,8}        58: {1,10}
%e A364058     21: {2,4}      39: {2,6}        59: {17}
%e A364058     22: {1,5}      41: {13}         60: {1,1,2,3}
%t A364058 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A364058 Select[Range[100],Median[prifacs[#]]>2&]
%Y A364058 For mean instead of median we have A057716, counted by A000065.
%Y A364058 These partitions are counted by A238495.
%Y A364058 The complement is A364056, counted by A027336, low version A363488.
%Y A364058 A000975 counts subsets with integer median, A051293 for mean.
%Y A364058 A124943 counts partitions by low median, high version A124944.
%Y A364058 A360005 gives twice the median of prime indices, A360459 for prime factors.
%Y A364058 A359893 and A359901 count partitions by median.
%Y A364058 Cf. A002865, A316413, A325347, A327473, A327476, A363727.
%K A364058 nonn
%O A364058 1,1
%A A364058 _Gus Wiseman_, Jul 14 2023