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Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.
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%I #16 Jan 08 2021 21:17:10

%S 0,1,1,1,1,2,1,1,2,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,2,2,2,2,1,3,1,1,2,2,

%T 2,2,1,2,2,2,1,3,1,2,3,2,1,2,2,3,2,2,1,2,2,2,2,2,1,3,1,2,3,1,2,3,1,2,

%U 2,3,1,2,1,2,3,2,2,3,1,2,2,2,1,3,2,2,2

%N Origin-to-boundary graph-distance of the Young diagram of the integer partition with Heinz number n.

%C The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

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

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000783">St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram</a>

%F A257990(n) <= a(n) <= 2 * A257990(n).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];

%t Table[otb[Reverse[primeMS[n]]],{n,100}]

%Y Cf. A001221, A001222, A056239, A061395, A065770, A112798, A115994, A174090, A257990, A297113, A325166, A325167, A325170.

%K nonn

%O 1,6

%A _Gus Wiseman_, Apr 05 2019