A325186 - OEIS

%I #5 Feb 16 2025 08:33:58

%S 3,4,15,18,21,25,27,33,36,39,51,57,69,72,87,93,105,111,123,129,141,

%T 144,147,150,159,165,175,177,183,195,201,213,219,225,231,237,245,249,

%U 250,255,267,273,275,285,288,291,300,303,309,321,325,327,339,343,345,357

%N Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.

%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.

%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="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.

%H Gus Wiseman, <a href="/A325186/a325186.png">Young diagrams for the first 25 terms</a>.

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

%e     3: {2}

%e     4: {1,1}

%e    15: {2,3}

%e    18: {1,2,2}

%e    21: {2,4}

%e    25: {3,3}

%e    27: {2,2,2}

%e    33: {2,5}

%e    36: {1,1,2,2}

%e    39: {2,6}

%e    51: {2,7}

%e    57: {2,8}

%e    69: {2,9}

%e    72: {1,1,1,2,2}

%e    87: {2,10}

%e    93: {2,11}

%e   105: {2,3,4}

%e   111: {2,12}

%e   123: {2,13}

%e   129: {2,14}

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

%t ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];

%t corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];

%t Select[Range[100],Apply[Plus,If[#==1,{},FixedPointList[corpos,ptnmat[primeptn[#]]][[-3]]],{0,1}]==2&]

%Y Cf. A006918, A056239, A065770, A112798.

%Y Cf. A325164, A325169, A325170, A325183, A325184, A325185, A325190, A325197.

%K nonn,changed

%O 1,1

%A _Gus Wiseman_, Apr 08 2019