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

1,1

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.

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.

Example

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}

Mathematica

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

Crossrefs

Positions of numbers > 2 in A257990.

Cf. A006918, A056239, A084835, A112798, A115994, A117485, A252464, A325163, A325170.

Sequence in context: A069656 A196943 A307386 * A038513 A251125 A252065

Adjacent sequences: A307512 A307513 A307514 * A307516 A307517 A307518

Keyword

nonn

Author

Gus Wiseman, Apr 12 2019

Status

approved