A349150 Heinz numbers of integer partitions with at most one odd part.

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109

Offset: 1

Author

Gus Wiseman, Nov 10 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.

Also Heinz numbers of partitions with conjugate alternating sum <= 1.

Examples

			The terms and their prime indices begin:
      1: {}          23: {9}         49: {4,4}
      2: {1}         26: {1,6}       51: {2,7}
      3: {2}         27: {2,2,2}     53: {16}
      5: {3}         29: {10}        54: {1,2,2,2}
      6: {1,2}       31: {11}        57: {2,8}
      7: {4}         33: {2,5}       58: {1,10}
      9: {2,2}       35: {3,4}       59: {17}
     11: {5}         37: {12}        61: {18}
     13: {6}         38: {1,8}       63: {2,2,4}
     14: {1,4}       39: {2,6}       65: {3,6}
     15: {2,3}       41: {13}        67: {19}
     17: {7}         42: {1,2,4}     69: {2,9}
     18: {1,2,2}     43: {14}        71: {20}
     19: {8}         45: {2,2,3}     73: {21}
     21: {2,4}       47: {15}        74: {1,12}
		

Crossrefs

The case of no odd parts is A066207, counted by A000041 up to 0's.

Requiring all odd parts gives A066208, counted by A000009.

These partitions are counted by A100824, even-length case A349149.

These are the positions of 0's and 1's in A257991.

The conjugate partitions are ranked by A349151.

The case of one odd part is A349158, counted by A000070 up to 0's.

A056239 adds up prime indices, row sums of A112798.

A122111 is a representation of partition conjugation.

A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.

A316524 gives the alternating sum of prime indices (reverse: A344616).

A325698 ranks partitions with as many even as odd parts, counted by A045931.

A340932 ranks partitions whose least part is odd, counted by A026804.

A345958 ranks partitions with alternating sum 1.

A349157 ranks partitions with as many even parts as odd conjugate parts.

Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[Reverse[primeMS[#]],_?OddQ]<=1&]

Formula

Union of A066207 (no odd parts) and A349158 (one odd part).