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

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86

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

Links

Table of n, a(n) for n=1..66.

Formula

A360005(a(n)) > 1.

A360459(a(n)) > 2.

Example

The terms together with their prime indices begin:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

Mathematica

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100], Median[prifacs[#]]>2&]

Crossrefs

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

These partitions are counted by A238495.

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

A000975 counts subsets with integer median, A051293 for mean.

A124943 counts partitions by low median, high version A124944.

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

A359893 and A359901 count partitions by median.

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

Sequence in context: A116883 A256543 A186145 * A335740 A352873 A047984

Adjacent sequences: A364055 A364056 A364057 * A364059 A364060 A364061

Keyword

nonn

Author

Gus Wiseman, Jul 14 2023

Status

approved