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%I #8 Dec 28 2022 09:05:02
%S 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,
%T 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,
%U 7,5,0,2,0,11,2,7,1,6,0,2,0,12,0,4,4,13
%N 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.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F a(n) = A056239(n) - A001222(n) * A055396(n).
%F a(n) = A056239(n) - A359360(n).
%e 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.
%e . . . o o o
%e . . . o o
%e . . . o o
%e . . .
%e The size of the complement is 7, so a(7865) = 7.
%t 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}]
%Y The opposite version is A326844.
%Y Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.
%Y A055396 gives minimum prime index, maximum A061395.
%Y A112798 list prime indices, sum A056239.
%Y A243055 subtracts the least prime index from the greatest.
%Y A326846 = size of the smallest rectangle containing the prime indices of n.
%Y A358195 gives Heinz numbers of rows of A358172, even bisection A241916.
%Y Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.
%K nonn
%O 1,10
%A _Gus Wiseman_, Dec 27 2022
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