A339658 - OEIS

A339658

Heinz numbers of loop-graphical partitions (of even numbers).

%I #11 Jan 05 2021 14:57:48

%S 1,3,4,9,10,12,16,25,27,28,30,36,40,48,63,64,70,75,81,84,88,90,100,

%T 108,112,120,144,147,160,175,189,192,196,198,208,210,220,225,243,250,

%U 252,256,264,270,280,300,324,336,343,352,360,400,432,441,448,462,468,480

%N Heinz numbers of loop-graphical partitions (of even numbers).

%C Equals the image of A181819 applied to the set of terms of A320912.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%C A partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656.

%C The following are equivalent characteristics for any positive integer n:

%C (1) the prime factors of n can be partitioned into distinct pairs;

%C (2) n can be factored into distinct semiprimes;

%C (3) the prime signature of n is loop-graphical.

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

%F A300061 = A339657 \/ A339658.

%e The sequence of terms > 1 together with their prime indices begins:

%e 3: {2} 70: {1,3,4} 192: {1,1,1,1,1,1,2}

%e 4: {1,1} 75: {2,3,3} 196: {1,1,4,4}

%e 9: {2,2} 81: {2,2,2,2} 198: {1,2,2,5}

%e 10: {1,3} 84: {1,1,2,4} 208: {1,1,1,1,6}

%e 12: {1,1,2} 88: {1,1,1,5} 210: {1,2,3,4}

%e 16: {1,1,1,1} 90: {1,2,2,3} 220: {1,1,3,5}

%e 25: {3,3} 100: {1,1,3,3} 225: {2,2,3,3}

%e 27: {2,2,2} 108: {1,1,2,2,2} 243: {2,2,2,2,2}

%e 28: {1,1,4} 112: {1,1,1,1,4} 250: {1,3,3,3}

%e 30: {1,2,3} 120: {1,1,1,2,3} 252: {1,1,2,2,4}

%e 36: {1,1,2,2} 144: {1,1,1,1,2,2} 256: {1,1,1,1,1,1,1,1}

%e 40: {1,1,1,3} 147: {2,4,4} 264: {1,1,1,2,5}

%e 48: {1,1,1,1,2} 160: {1,1,1,1,1,3} 270: {1,2,2,2,3}

%e 63: {2,2,4} 175: {3,3,4} 280: {1,1,1,3,4}

%e 64: {1,1,1,1,1,1} 189: {2,2,2,4} 300: {1,1,2,3,3}

%e For example, the four loop-graphs with degrees y = (3,1,1,1) are:

%e {{1,1},{1,2},{3,4}}

%e {{1,1},{1,3},{2,4}}

%e {{1,1},{1,4},{2,3}}

%e {{1,2},{1,3},{1,4}},

%e so the Heinz number 40 is in the sequence. On the other hand, the three loop-multigraphs with degrees y = (4,4) are

%e {{1,1},{1,1},{2,2},{2,2}}

%e {{1,1},{1,2},{1,2},{2,2}}

%e {{1,2},{1,2},{1,2},{1,2}},

%e but none of these is a loop-graph, so the Heinz number 49 is not in the sequence.

%t spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];

%t mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[25],Select[mpsbin[nrmptn[#]],UnsameQ@@#&]!={}&]

%Y A320912 has these prime shadows (see A181819).

%Y A339656 counts these partitions.

%Y A339657 ranks the complement, counted by A339655.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A101048 counts partitions into semiprimes.

%Y A320655 counts factorizations into semiprimes.

%Y The following count vertex-degree partitions and give their Heinz numbers:

%Y - A058696 counts partitions of 2n (A300061).

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A000569 counts graphical partitions (A320922).

%Y The following count partitions of even length and give their Heinz numbers:

%Y - A027187 has no additional conditions (A028260).

%Y - A338914 can be partitioned into strict pairs (A320911).

%Y - A338916 can be partitioned into distinct pairs (A320912).

%Y - A339560 can be partitioned into distinct strict pairs (A339561).

%Y Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A320892, A338898, A338912, A338913, A339112.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 18 2020