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A359358
Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.
5
0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 2, 0, 4, 0, 4, 1, 8, 0, 1, 0, 4, 5, 5, 0, 3, 2, 3, 6, 9, 0, 3, 0, 10, 2, 0, 3, 5, 0, 6, 7, 5, 0, 2, 0, 11, 2, 7, 1, 6, 0, 2, 0, 12, 0, 4, 4, 13
OFFSET
1,10
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
a(n) = A056239(n) - A001222(n) * A055396(n).
a(n) = A056239(n) - A359360(n).
EXAMPLE
The partition with Heinz number 7865 is (6,5,5,3), which has the following diagram. The 3 X 4 rectangle is shown in dots.
. . . o o o
. . . o o
. . . o o
. . .
The size of the complement is 7, so a(7865) = 7.
MATHEMATICA
Table[If[n==1, 0, With[{q=Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]}, Total[q]-q[[1]]*Length[q]]], {n, 100}]
CROSSREFS
The opposite version is A326844.
Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
A055396 gives minimum prime index, maximum A061395.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
A326846 = size of the smallest rectangle containing the prime indices of n.
A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
Sequence in context: A242411 A286470 A243055 * A318371 A363157 A363952
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 27 2022
STATUS
approved