cp's OEIS Frontend

A325186 Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.

%I A325186 #5 Feb 16 2025 08:33:58
%S A325186 3,4,15,18,21,25,27,33,36,39,51,57,69,72,87,93,105,111,123,129,141,
%T A325186 144,147,150,159,165,175,177,183,195,201,213,219,225,231,237,245,249,
%U A325186 250,255,267,273,275,285,288,291,300,303,309,321,325,327,339,343,345,357
%N A325186 Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.
%C A325186 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 A325186 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325186 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.
%H A325186 Gus Wiseman, <a href="/A325186/a325186.png">Young diagrams for the first 25 terms</a>.
%e A325186 The sequence of terms together with their prime indices begins:
%e A325186     3: {2}
%e A325186     4: {1,1}
%e A325186    15: {2,3}
%e A325186    18: {1,2,2}
%e A325186    21: {2,4}
%e A325186    25: {3,3}
%e A325186    27: {2,2,2}
%e A325186    33: {2,5}
%e A325186    36: {1,1,2,2}
%e A325186    39: {2,6}
%e A325186    51: {2,7}
%e A325186    57: {2,8}
%e A325186    69: {2,9}
%e A325186    72: {1,1,1,2,2}
%e A325186    87: {2,10}
%e A325186    93: {2,11}
%e A325186   105: {2,3,4}
%e A325186   111: {2,12}
%e A325186   123: {2,13}
%e A325186   129: {2,14}
%t A325186 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325186 ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];
%t A325186 corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];
%t A325186 Select[Range[100],Apply[Plus,If[#==1,{},FixedPointList[corpos,ptnmat[primeptn[#]]][[-3]]],{0,1}]==2&]
%Y A325186 Cf. A006918, A056239, A065770, A112798.
%Y A325186 Cf. A325164, A325169, A325170, A325183, A325184, A325185, A325190, A325197.
%K A325186 nonn
%O A325186 1,1
%A A325186 _Gus Wiseman_, Apr 08 2019