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

%I A307386 #6 Apr 07 2019 00:03:12
%S A307386 125,175,245,250,275,325,343,350,375,385,425,455,475,490,500,525,539,
%T A307386 550,575,595,605,625,637,650,665,686,700,715,725,735,750,770,775,805,
%U A307386 825,833,845,847,850,875,910,925,931,935,950,975,980,1000,1001,1015,1025
%N A307386 Heinz numbers of integer partitions with Durfee square of length 3.
%C A307386 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A307386 The Durfee square of an integer partition is the largest square contained in its Young diagram.
%H A307386 Gus Wiseman, <a href="/A307386/a307386.png">Young diagrams corresponding to the first 36 terms</a>.
%e A307386 The sequence of terms together with their prime indices begins:
%e A307386   125: {3,3,3}
%e A307386   175: {3,3,4}
%e A307386   245: {3,4,4}
%e A307386   250: {1,3,3,3}
%e A307386   275: {3,3,5}
%e A307386   325: {3,3,6}
%e A307386   343: {4,4,4}
%e A307386   350: {1,3,3,4}
%e A307386   375: {2,3,3,3}
%e A307386   385: {3,4,5}
%e A307386   425: {3,3,7}
%e A307386   455: {3,4,6}
%e A307386   475: {3,3,8}
%e A307386   490: {1,3,4,4}
%e A307386   500: {1,1,3,3,3}
%e A307386   525: {2,3,3,4}
%e A307386   539: {4,4,5}
%e A307386   550: {1,3,3,5}
%e A307386   575: {3,3,9}
%e A307386   595: {3,4,7}
%t A307386 durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
%t A307386 Select[Range[100],durf[#]==3&]
%Y A307386 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.
%Y A307386 Cf. A056239, A112798, A115994, A252464, A325163, A325170.
%K A307386 nonn
%O A307386 1,1
%A A307386 _Gus Wiseman_, Apr 06 2019