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

%I A307515 #7 Apr 12 2019 08:26:32
%S A307515 125,175,245,250,275,325,343,350,375,385,425,455,475,490,500,525,539,
%T A307515 550,575,595,605,625,637,650,665,686,700,715,725,735,750,770,775,805,
%U A307515 825,833,845,847,850,875,910,925,931,935,950,975,980,1000,1001,1015,1025
%N A307515 Heinz numbers of integer partitions with Durfee square of length > 2.
%C A307515 First differs from A307386 in having 7^4 = 2401.
%C A307515 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A307515 The Durfee square of an integer partition is the largest square contained in its Young diagram.
%C A307515 The enumeration of these partitions by sum is given by A084835.
%D A307515 Richard P. Stanley, Enumerative Combinatorics, Volume 2,  Cambridge University Press, 1999, p. 289.
%H A307515 Gus Wiseman, <a href="/A307515/b307515.txt">Table of n, a(n) for n = 1..22485</a>
%H A307515 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000183">St000183: The side length of the Durfee square of an integer partition</a>
%H A307515 Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.
%e A307515 The sequence of terms together with their prime indices begins:
%e A307515   125: {3,3,3}
%e A307515   175: {3,3,4}
%e A307515   245: {3,4,4}
%e A307515   250: {1,3,3,3}
%e A307515   275: {3,3,5}
%e A307515   325: {3,3,6}
%e A307515   343: {4,4,4}
%e A307515   350: {1,3,3,4}
%e A307515   375: {2,3,3,3}
%e A307515   385: {3,4,5}
%e A307515   425: {3,3,7}
%e A307515   455: {3,4,6}
%e A307515   475: {3,3,8}
%e A307515   490: {1,3,4,4}
%e A307515   500: {1,1,3,3,3}
%e A307515   525: {2,3,3,4}
%e A307515   539: {4,4,5}
%e A307515   550: {1,3,3,5}
%e A307515   575: {3,3,9}
%e A307515   595: {3,4,7}
%t A307515 durf[n_]:=Length[Select[Range[PrimeOmega[n]], Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]][[#]]>=#&]];
%t A307515 Select[Range[100], durf[#]>2&]
%Y A307515 Positions of numbers > 2 in A257990.
%Y A307515 Cf. A006918, A056239, A084835, A112798, A115994, A117485, A252464, A325163, A325170.
%K A307515 nonn
%O A307515 1,1
%A A307515 _Gus Wiseman_, Apr 12 2019