login
Heinz numbers of integer partitions with Durfee square of length 2.
17

%I #6 Apr 05 2019 09:29:18

%S 9,15,18,21,25,27,30,33,35,36,39,42,45,49,50,51,54,55,57,60,63,65,66,

%T 69,70,72,75,77,78,81,84,85,87,90,91,93,95,98,99,100,102,105,108,110,

%U 111,114,115,117,119,120,121,123,126,129,130,132,133,135,138,140

%N Heinz numbers of integer partitions with Durfee square of length 2.

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

%C Also positions of 2 in A257990.

%C First differs from A105441 in lacking 125.

%C 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.

%H Gus Wiseman, <a href="/A325164/a325164.png">Young diagrams corresponding to the first 36 terms.</a>

%e The sequence of terms together with their prime indices begins:

%e 9: {2,2}

%e 15: {2,3}

%e 18: {1,2,2}

%e 21: {2,4}

%e 25: {3,3}

%e 27: {2,2,2}

%e 30: {1,2,3}

%e 33: {2,5}

%e 35: {3,4}

%e 36: {1,1,2,2}

%e 39: {2,6}

%e 42: {1,2,4}

%e 45: {2,2,3}

%e 49: {4,4}

%e 50: {1,3,3}

%e 51: {2,7}

%e 54: {1,2,2,2}

%e 55: {3,5}

%e 57: {2,8}

%e 60: {1,1,2,3}

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

%t Select[Range[100],durf[#]==2&]

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 05 2019