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Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.
6

%I #6 Apr 17 2019 19:09:13

%S 2,4,6,12,18,20,24,28,40,48,60,84,90,120,126,132,140,150,156,168,180,

%T 198,204,220,228,234,240,252,260,264,270,276,280

%N Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

%C The enumeration of these partitions by sum is given by A325254.

%C The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

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

%e The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:

%e 2: {1} (1)

%e 4: {1,1} (2,1)

%e 6: {1,2} (2,2,1)

%e 12: {1,1,2} (3,2,2,1)

%e 18: {1,2,2} (3,2,2,1)

%e 20: {1,1,3} (3,2,2,1)

%e 24: {1,1,1,2} (4,2,2,1)

%e 28: {1,1,4} (3,2,2,1)

%e 40: {1,1,1,3} (4,2,2,1)

%e 48: {1,1,1,1,2} (5,2,2,1)

%e 60: {1,1,2,3} (4,3,2,2,1)

%e 84: {1,1,2,4} (4,3,2,2,1)

%e 90: {1,2,2,3} (4,3,2,2,1)

%e 120: {1,1,1,2,3} (5,3,2,2,1)

%e 126: {1,2,2,4} (4,3,2,2,1)

%e 132: {1,1,2,5} (4,3,2,2,1)

%e 140: {1,1,3,4} (4,3,2,2,1)

%e 150: {1,2,3,3} (4,3,2,2,1)

%e 156: {1,1,2,6} (4,3,2,2,1)

%e 168: {1,1,1,2,4} (5,3,2,2,1)

%e 180: {1,1,2,2,3} (5,3,2,2,1)

%t nn=30;

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

%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];

%t mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];

%t Select[Range[Prime[nn]],fdadj[primeMS[#]]==mfds[[Total[primeMS[#]]]]&]

%Y Cf. A011784, A056239, A112798, A118914, A181819, A182857, A225486, A323014, A323023, A325254, A325258, A325277, A325278, A325282.

%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

%K nonn,more

%O 1,1

%A _Gus Wiseman_, Apr 17 2019