A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Author

Gus Wiseman, May 14 2023

Keywords

• nonn

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.

Examples

			The terms together with their prime indices begin:
    10: {1,3}        44: {1,1,5}      70: {1,3,4}
    14: {1,4}        46: {1,9}        74: {1,12}
    20: {1,1,3}      50: {1,3,3}      76: {1,1,8}
    22: {1,5}        51: {2,7}        78: {1,2,6}
    26: {1,6}        52: {1,1,6}      80: {1,1,1,1,3}
    28: {1,1,4}      56: {1,1,1,4}    82: {1,13}
    30: {1,2,3}      57: {2,8}        84: {1,1,2,4}
    33: {2,5}        58: {1,10}       85: {3,7}
    34: {1,7}        60: {1,1,2,3}    86: {1,14}
    38: {1,8}        62: {1,11}       87: {2,10}
    39: {2,6}        66: {1,2,5}      88: {1,1,1,5}
    40: {1,1,1,3}    68: {1,1,7}      90: {1,2,2,3}
    42: {1,2,4}      69: {2,9}        92: {1,1,9}
		

Crossrefs

For prime factors instead of indices we have A069900, complement A081306.

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

Partitions of this type are counted by A237820.

The complement is A362981, counted by A237824.

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

Programs

Mathematica

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