A325164 Heinz numbers of integer partitions with Durfee square of length 2.

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 126, 129, 130, 132, 133, 135, 138, 140

Offset: 1

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

Comments

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

Also positions of 2 in A257990.

First differs from A105441 in lacking 125.

The Durfee length 1 case is A093641. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

Examples

			The sequence of terms together with their prime indices begins:
   9: {2,2}
  15: {2,3}
  18: {1,2,2}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  60: {1,1,2,3}
		

Links

• Gus Wiseman, Young diagrams corresponding to the first 36 terms.

Crossrefs

Cf. A006918, A056239, A093641, A112798, A115994, A117485, A252464, A257990, A325163, A325170.

Programs

Mathematica

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
Select[Range[100],durf[#]==2&]