A358102 Numbers of the form prime(w)*prime(x)*prime(y) with w >= x >= y such that 2w = 3x + 4y.

66, 153, 266, 609, 806, 1295, 1599, 1634, 2107, 3021, 3055, 3422, 5254, 5369, 5795, 5829, 7138, 8769, 9443, 9581, 10585, 10706, 12337, 12513, 13298, 16465, 16511, 16849, 17013, 18602, 21983, 22145, 23241, 23542, 26159, 29014, 29607, 29945, 30943, 32623, 32809

Offset: 1

Author

Gus Wiseman, Nov 02 2022

Keywords

• nonn

Comments

Also Heinz numbers of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The terms together with their prime indices begin:
     66: {1,2,5}
    153: {2,2,7}
    266: {1,4,8}
    609: {2,4,10}
    806: {1,6,11}
   1295: {3,4,12}
   1599: {2,6,13}
   1634: {1,8,14}
   2107: {4,4,14}
   3021: {2,8,16}
   3055: {3,6,15}
   3422: {1,10,17}
   5254: {1,12,20}
   5369: {4,6,17}
   5795: {3,8,18}
   5829: {2,10,19}
   7138: {1,14,23}
   8769: {2,12,22}
		

Crossrefs

The ordered version is A357489, apparently counted by A008676.

These partitions are counted by A357849.

A000040 lists the primes.

A000041 counts partitions, strict A000009.

A003963 multiplies prime indices.

A056239 adds up prime indices.

Cf. A000720, A001221, A001222, A215366, A296150, A318283.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],PrimeOmega[#]==3&&2*primeMS[#][[-1]]==3*primeMS[#][[-2]]+4*primeMS[#][[-3]]&]