A325363 Heinz numbers of integer partitions into nonzero triangular numbers A000217.

1, 2, 4, 5, 8, 10, 13, 16, 20, 25, 26, 29, 32, 40, 47, 50, 52, 58, 64, 65, 73, 80, 94, 100, 104, 107, 116, 125, 128, 130, 145, 146, 151, 160, 169, 188, 197, 200, 208, 214, 232, 235, 250, 256, 257, 260, 290, 292, 302, 317, 320, 325, 338, 365, 376, 377, 394, 397

Offset: 1

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A007294.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    5: {3}
    8: {1,1,1}
   10: {1,3}
   13: {6}
   16: {1,1,1,1}
   20: {1,1,3}
   25: {3,3}
   26: {1,6}
   29: {10}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   58: {1,10}
   64: {1,1,1,1,1,1}
   65: {3,6}
		

Crossrefs

Cf. A000217, A007294, A056239, A112798, A240026, A325327, A325360, A325362, A325367, A325390, A325394, A325400.

Programs

Mathematica

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
trgs=Table[n*(n+1)/2,{n,Sqrt[2*PrimePi[nn]]}];
Select[Range[nn],SubsetQ[trgs,primeMS[#]]&]