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A325185 Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

%I A325185 #7 Feb 16 2025 08:33:58
%S A325185 2,6,9,10,12,14,20,22,24,26,28,30,34,38,40,42,44,45,46,48,50,52,56,58,
%T A325185 60,62,63,66,68,70,74,75,76,78,80,82,84,86,88,90,92,94,96,98,99,100,
%U A325185 102,104,106,110,112,114,116,117,118,120,122,124,125,126,130,132
%N A325185 Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.
%C A325185 The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary. The sequence gives all Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 1.
%C A325185 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325185 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%H A325185 Gus Wiseman, <a href="/A325185/a325185.png">Young diagrams for the first 25 terms</a>.
%e A325185 The sequence of terms together with their prime indices begins:
%e A325185     2: {1}
%e A325185     6: {1,2}
%e A325185     9: {2,2}
%e A325185    10: {1,3}
%e A325185    12: {1,1,2}
%e A325185    14: {1,4}
%e A325185    20: {1,1,3}
%e A325185    22: {1,5}
%e A325185    24: {1,1,1,2}
%e A325185    26: {1,6}
%e A325185    28: {1,1,4}
%e A325185    30: {1,2,3}
%e A325185    34: {1,7}
%e A325185    38: {1,8}
%e A325185    40: {1,1,1,3}
%e A325185    42: {1,2,4}
%e A325185    44: {1,1,5}
%e A325185    45: {2,2,3}
%e A325185    46: {1,9}
%e A325185    48: {1,1,1,1,2}
%t A325185 hptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325185 otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325185 Select[Range[2,100],otb[hptn[#]]>otb[Rest[hptn[#]]]&&otb[hptn[#]]>otb[DeleteCases[hptn[#]-1,0]]&]
%Y A325185 Cf. A001222, A056239, A061395, A065770, A112798, A188674.
%Y A325185 Cf. A325169, A325183, A325184, A325186, A325187, A325196.
%K A325185 nonn
%O A325185 1,1
%A A325185 _Gus Wiseman_, Apr 08 2019