A307515 Heinz numbers of integer partitions with Durfee square of length > 2.

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 12 2019

Keywords

• nonn

Comments

First differs from A307386 in having 7^4 = 2401.

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.

The enumeration of these partitions by sum is given by A084835.

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}
		

References

• Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Links

• Gus Wiseman, Table of n, a(n) for n = 1..22485
• FindStat, St000183: The side length of the Durfee square of an integer partition
• Wikipedia, Durfee square.

Crossrefs

Positions of numbers > 2 in A257990.

Cf. A006918, A056239, A084835, A112798, A115994, A117485, 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[#]>2&]