A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 105, 107, 108, 109, 113, 119, 121, 125

Offset

1,2

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.

By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.

Links

Table of n, a(n) for n=1..65.

Example

The terms together with their prime indices begin:
     1: {}         16: {1,1,1,1}      36: {1,1,2,2}
     2: {1}        17: {7}            37: {12}
     3: {2}        18: {1,2,2}        41: {13}
     4: {1,1}      19: {8}            43: {14}
     5: {3}        21: {2,4}          45: {2,2,3}
     6: {1,2}      23: {9}            47: {15}
     7: {4}        24: {1,1,1,2}      48: {1,1,1,1,2}
     8: {1,1,1}    25: {3,3}          49: {4,4}
     9: {2,2}      27: {2,2,2}        53: {16}
    11: {5}        29: {10}           54: {1,2,2,2}
    12: {1,1,2}    31: {11}           55: {3,5}
    13: {6}        32: {1,1,1,1,1}    59: {17}
    15: {2,3}      35: {3,4}          61: {18}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], 2*Min@@prix[#]>=Max@@prix[#]&]

Crossrefs

For prime factors instead of indices we have A081306.

Prime indices are listed by A112798, length A001222, sum A056239.

The complement is A362982, counted by A237820.

Partitions of this type are counted by A237824.

Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

Sequence in context: A253784 A342191 A251726 * A193671 A073491 A066311

Adjacent sequences: A362978 A362979 A362980 * A362982 A362983 A362984

Keyword

nonn

Author

Gus Wiseman, May 14 2023

Status

approved