A325185 - OEIS

%I #6 Apr 10 2019 22:01:46

%S 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 60,62,63,66,68,70,74,75,76,78,80,82,84,86,88,90,92,94,96,98,99,100,

%U 102,104,106,110,112,114,116,117,118,120,122,124,125,126,130,132

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.

%H Gus Wiseman, <a href="/A325185/a325185.png">Young diagrams for the first 25 terms</a>.

%e The sequence of terms together with their prime indices begins:

%e     2: {1}

%e     6: {1,2}

%e     9: {2,2}

%e    10: {1,3}

%e    12: {1,1,2}

%e    14: {1,4}

%e    20: {1,1,3}

%e    22: {1,5}

%e    24: {1,1,1,2}

%e    26: {1,6}

%e    28: {1,1,4}

%e    30: {1,2,3}

%e    34: {1,7}

%e    38: {1,8}

%e    40: {1,1,1,3}

%e    42: {1,2,4}

%e    44: {1,1,5}

%e    45: {2,2,3}

%e    46: {1,9}

%e    48: {1,1,1,1,2}

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

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

%t Select[Range[2,100],otb[hptn[#]]>otb[Rest[hptn[#]]]&&otb[hptn[#]]>otb[DeleteCases[hptn[#]-1,0]]&]

%Y Cf. A001222, A056239, A061395, A065770, A112798, A188674.

%Y Cf. A325169, A325183, A325184, A325186, A325187, A325196.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 08 2019