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

1,1

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.

Links

Table of n, a(n) for n=1..60.

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

Example

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}

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

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

Sequence in context: A358576 A110473 A105441 * A093642 A364566 A177733

Adjacent sequences: A325161 A325162 A325163 * A325165 A325166 A325167

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Status

approved