A307515 - OEIS

%I #7 Apr 12 2019 08:26:32

%S 125,175,245,250,275,325,343,350,375,385,425,455,475,490,500,525,539,

%T 550,575,595,605,625,637,650,665,686,700,715,725,735,750,770,775,805,

%U 825,833,845,847,850,875,910,925,931,935,950,975,980,1000,1001,1015,1025

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

%C First differs from A307386 in having 7^4 = 2401.

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

%C The Durfee square of an integer partition is the largest square contained in its Young diagram.

%C The enumeration of these partitions by sum is given by A084835.

%D Richard P. Stanley, Enumerative Combinatorics, Volume 2,  Cambridge University Press, 1999, p. 289.

%H Gus Wiseman, <a href="/A307515/b307515.txt">Table of n, a(n) for n = 1..22485</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000183">St000183: The side length of the Durfee square of an integer partition</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Durfee_square">Durfee square</a>.

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

%e   125: {3,3,3}

%e   175: {3,3,4}

%e   245: {3,4,4}

%e   250: {1,3,3,3}

%e   275: {3,3,5}

%e   325: {3,3,6}

%e   343: {4,4,4}

%e   350: {1,3,3,4}

%e   375: {2,3,3,3}

%e   385: {3,4,5}

%e   425: {3,3,7}

%e   455: {3,4,6}

%e   475: {3,3,8}

%e   490: {1,3,4,4}

%e   500: {1,1,3,3,3}

%e   525: {2,3,3,4}

%e   539: {4,4,5}

%e   550: {1,3,3,5}

%e   575: {3,3,9}

%e   595: {3,4,7}

%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 Positions of numbers > 2 in A257990.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 12 2019