A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

1, 4, 9, 25, 36, 48, 49, 100, 121, 160, 169, 196, 225, 289, 361, 441, 448, 484, 529, 567, 676, 750, 810, 841, 900, 961, 1080, 1089, 1156, 1200, 1225, 1369, 1408, 1440, 1444, 1521, 1681, 1764, 1849, 1920, 2116, 2209, 2268, 2352, 2601, 2809, 3024, 3025, 3159

Offset: 1

Author

Gus Wiseman, Aug 23 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 prime indices of 750 are {1,2,3,3,3}, with sum 12, while the distinct prime indices {1,2,3} have sum 6, so 750 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    25: {3,3}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    49: {4,4}
   100: {1,1,3,3}
   121: {5,5}
   160: {1,1,1,1,1,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   289: {7,7}
   361: {8,8}
   441: {2,2,4,4}
   448: {1,1,1,1,1,1,4}
		

Crossrefs

The LHS is A056239 (sum of prime indices).

The RHS is twice A066328.

Partitions of this type are counted by A364910.

A000041 counts integer partitions, strict A000009.

A001222 counts prime indices, distinct A001221.

A112798 lists prime indices, distinct A304038.

A116861 counts partitions by sum and sum of distinct parts.

A323092 counts double-free partitions, ranks A320340.

Cf. A364350, A364839, A364906, A364907, A364911, A364916.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[prix[#]]==2*Total[Union[prix[#]]]&]

Formula

A056239(a(n)) = 2*A066328(a(n)).