A325040 Heinz numbers of integer partitions with the same product of parts as their conjugate.

1, 2, 6, 9, 20, 30, 49, 56, 70, 75, 81, 84, 90, 125, 176, 182, 210, 264, 350, 416, 441, 532, 540, 546, 624, 660, 735, 910, 1088, 1100, 1260, 1378, 1386, 1443, 1520, 1560, 1624, 1632, 1715, 2100, 2310, 2401, 2405, 2432, 2489, 2600, 3024, 3267, 3276, 3648, 3744

Offset: 1

Author

Gus Wiseman, Mar 25 2019

Keywords

• nonn

Comments

For example, 182 is the Heinz number of (6,4,1) with product 24 and conjugate (3,2,2,2,1,1) with product also 24.

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 A325039.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   20: {1,1,3}
   30: {1,2,3}
   49: {4,4}
   56: {1,1,1,4}
   70: {1,3,4}
   75: {2,3,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
  125: {3,3,3}
  176: {1,1,1,1,5}
  182: {1,4,6}
  210: {1,2,3,4}
  264: {1,1,1,2,5}
  350: {1,3,3,4}
  416: {1,1,1,1,1,6}
		

Crossrefs

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A122111, A321650.

Cf. A325039, A325045.

Programs

Mathematica

priptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Times@@priptn[#]==Times@@conj[priptn[#]]&]