A307386 Heinz numbers of integer partitions with Durfee square of length 3.

125, 175, 245, 250, 275, 325, 343, 350, 375, 385, 425, 455, 475, 490, 500, 525, 539, 550, 575, 595, 605, 625, 637, 650, 665, 686, 700, 715, 725, 735, 750, 770, 775, 805, 825, 833, 845, 847, 850, 875, 910, 925, 931, 935, 950, 975, 980, 1000, 1001, 1015, 1025

Offset: 1

Author

Gus Wiseman, Apr 06 2019

Keywords

• nonn

Comments

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

The Durfee square of an integer partition is the largest square contained in its Young diagram.

Examples

			The sequence of terms together with their prime indices begins:
  125: {3,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  275: {3,3,5}
  325: {3,3,6}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  385: {3,4,5}
  425: {3,3,7}
  455: {3,4,6}
  475: {3,3,8}
  490: {1,3,4,4}
  500: {1,1,3,3,3}
  525: {2,3,3,4}
  539: {4,4,5}
  550: {1,3,3,5}
  575: {3,3,9}
  595: {3,4,7}
		

Links

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

Crossrefs

Positions of 3 in A257990. The Durfee length 1 case is A093641. The Durfee length 2 case is A325164. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

Cf. A056239, A112798, A115994, A252464, 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[#]==3&]