proposed

approved

editing

proposed

Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362612, A362616, A362619, A362621, A362622.

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

A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

A362611 counts modes in prime factorization, triangle version A362614.

A362613 counts co-modes in prime factorization, triangle version A362615.

Cf. `A002865, A027746, A053263, A171979, A237821, `A240302, A327473, A327476, `A362612, A362616, A362619, `A362620, A362621, A362622.

allocated for Gus WisemanHeinz 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

1,2

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.

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}

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], 2*Min@@prix[#]>=Max@@prix[#]&]

For prime factors instead of indices we have A081306.

The complement is A362982, counted by A237820.

Partitions of this type are counted by A237824.

A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

A362611 counts modes in prime factorization, triangle version A362614.

A362613 counts co-modes in prime factorization, triangle version A362615.

Cf. `A002865, A053263, A171979, A237821, `A240302, A327473, A327476, `A362612, A362616, A362619, `A362620, A362621, A362622.

allocated

nonn

Gus Wiseman, May 14 2023

approved

editing

allocated for Gus Wiseman

allocated

approved