A364058 - OEIS

%I #7 Jul 15 2023 05:42:35

%S 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 35,36,37,38,39,41,42,43,45,46,47,49,50,51,53,54,55,57,58,59,60,61,62,

%U 63,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83,84,85,86

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

%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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%F A360005(a(n)) > 1.

%F A360459(a(n)) > 2.

%e The terms together with their prime indices begin:

%e      3: {2}        23: {9}          42: {1,2,4}

%e      5: {3}        25: {3,3}        43: {14}

%e      6: {1,2}      26: {1,6}        45: {2,2,3}

%e      7: {4}        27: {2,2,2}      46: {1,9}

%e      9: {2,2}      29: {10}         47: {15}

%e     10: {1,3}      30: {1,2,3}      49: {4,4}

%e     11: {5}        31: {11}         50: {1,3,3}

%e     13: {6}        33: {2,5}        51: {2,7}

%e     14: {1,4}      34: {1,7}        53: {16}

%e     15: {2,3}      35: {3,4}        54: {1,2,2,2}

%e     17: {7}        36: {1,1,2,2}    55: {3,5}

%e     18: {1,2,2}    37: {12}         57: {2,8}

%e     19: {8}        38: {1,8}        58: {1,10}

%e     21: {2,4}      39: {2,6}        59: {17}

%e     22: {1,5}      41: {13}         60: {1,1,2,3}

%t prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t Select[Range[100],Median[prifacs[#]]>2&]

%Y For mean instead of median we have A057716, counted by A000065.

%Y These partitions are counted by A238495.

%Y The complement is A364056, counted by A027336, low version A363488.

%Y A000975 counts subsets with integer median, A051293 for mean.

%Y A124943 counts partitions by low median, high version A124944.

%Y A360005 gives twice the median of prime indices, A360459 for prime factors.

%Y A359893 and A359901 count partitions by median.

%Y Cf. A002865, A316413, A325347, A327473, A327476, A363727.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 14 2023