login
A307515
Heinz numbers of integer partitions with Durfee square of length > 2.
2
125, 175, 245, 250, 275, 325, 343, 350, 375, 385, 425, 455, 475, 490, 500, 525, 539, 550, 575, 595, 605, 625, 637, 650, 665, 686, 700, 715, 725, 735, 750, 770, 775, 805, 825, 833, 845, 847, 850, 875, 910, 925, 931, 935, 950, 975, 980, 1000, 1001, 1015, 1025
OFFSET
1,1
COMMENTS
First differs from A307386 in having 7^4 = 2401.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The Durfee square of an integer partition is the largest square contained in its Young diagram.
The enumeration of these partitions by sum is given by A084835.
REFERENCES
Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.
EXAMPLE
The sequence of terms together with their prime indices begins:
125: {3,3,3}
175: {3,3,4}
245: {3,4,4}
250: {1,3,3,3}
275: {3,3,5}
325: {3,3,6}
343: {4,4,4}
350: {1,3,3,4}
375: {2,3,3,3}
385: {3,4,5}
425: {3,3,7}
455: {3,4,6}
475: {3,3,8}
490: {1,3,4,4}
500: {1,1,3,3,3}
525: {2,3,3,4}
539: {4,4,5}
550: {1,3,3,5}
575: {3,3,9}
595: {3,4,7}
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
Positions of numbers > 2 in A257990.
Sequence in context: A069656 A196943 A307386 * A038513 A251125 A252065
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 12 2019
STATUS
approved