cp's OEIS Frontend

A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

%I A325195 #14 Apr 24 2019 10:11:21
%S A325195 0,0,1,1,2,0,3,2,1,1,4,1,5,2,1,3,6,1,7,1,2,3,8,2,2,4,2,2,9,0,10,4,3,5,
%T A325195 2,2,11,6,4,2,12,1,13,3,1,7,14,3,3,1,5,4,15,2,3,2,6,8,16,1,17,9,1,5,4,
%U A325195 2,18,5,7,1,19,3,20,10,1,6,3,3,21,3,3,11
%N A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.
%C A325195 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325195 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000384">St000384: The maximal part of the shifted composition of an integer partition</a>
%H A325195 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>
%e A325195 The partition (3,3) has Heinz number 25 and diagram
%e A325195   o o o
%e A325195   o o o
%e A325195 containing maximal triangular partition
%e A325195   o o
%e A325195   o
%e A325195 and contained in minimal triangular partition
%e A325195   o o o o
%e A325195   o o o
%e A325195   o o
%e A325195   o
%e A325195 so a(25) = 4 - 2 = 2.
%t A325195 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325195 otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325195 otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325195 Table[otbmax[primeptn[n]]-otb[primeptn[n]],{n,100}]
%Y A325195 Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, A325199, A325200.
%K A325195 nonn
%O A325195 1,5
%A A325195 _Gus Wiseman_, Apr 11 2019