proposed

approved

editing

proposed

Cf. A000720, A001222, `A003963, ~A005087, `A257992, A318400, `A324927, A358137, ~A358195.

allocated for Gus WisemanHeinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 83, 84, 85, 86

1,1

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.

A257991(a(n)) > 0.

The terms together with their prime indices begin:

2: {1}

4: {1,1}

5: {3}

6: {1,2}

8: {1,1,1}

10: {1,3}

11: {5}

12: {1,1,2}

14: {1,4}

15: {2,3}

16: {1,1,1,1}

17: {7}

18: {1,2,2}

20: {1,1,3}

22: {1,5}

23: {9}

24: {1,1,1,2}

Select[Range[100], Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]

The complement is A066207, counted by A035363.

For all odd parts we have A066208, counted by A000009.

Partitions of this type are counted by A086543.

For even instead of odd we have A324929, counted by A047967.

A031368 lists primes of odd index.

A112798 list prime indices, sum A056239.

A257991 counts odd prime indices, distinct A324966.

Cf. A000720, A001222, `A003963, ~A005087, `A257992, A318400, `A324927, A358137, ~A358195.

allocated

nonn

Gus Wiseman, Oct 14 2023

approved

editing

allocated for Gus Wiseman

allocated

approved