A366322 Heinz 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

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.

Links

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

Formula

A257991(a(n)) > 0.

Example

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}

Mathematica

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

Crossrefs

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, A257992, A318400, A324927, A358137.

Sequence in context: A080653 A115836 A176554 * A284895 A285354 A050505

Adjacent sequences: A366319 A366320 A366321 * A366323 A366324 A366325

Keyword

nonn

Author

Gus Wiseman, Oct 14 2023

Status

approved