A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

2, 4, 6, 12, 18, 20, 24, 28, 40, 48, 60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280

Offset: 1

Author

Gus Wiseman, Apr 17 2019

Keywords

• nonn
• more

Comments

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

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).

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

Examples

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
  2:   {1}         (1)
  4:   {1,1}       (2,1)
  6:   {1,2}       (2,2,1)
  12:  {1,1,2}     (3,2,2,1)
  18:  {1,2,2}     (3,2,2,1)
  20:  {1,1,3}     (3,2,2,1)
  24:  {1,1,1,2}   (4,2,2,1)
  28:  {1,1,4}     (3,2,2,1)
  40:  {1,1,1,3}   (4,2,2,1)
  48:  {1,1,1,1,2} (5,2,2,1)
  60:  {1,1,2,3}   (4,3,2,2,1)
  84:  {1,1,2,4}   (4,3,2,2,1)
  90:  {1,2,2,3}   (4,3,2,2,1)
  120: {1,1,1,2,3} (5,3,2,2,1)
  126: {1,2,2,4}   (4,3,2,2,1)
  132: {1,1,2,5}   (4,3,2,2,1)
  140: {1,1,3,4}   (4,3,2,2,1)
  150: {1,2,3,3}   (4,3,2,2,1)
  156: {1,1,2,6}   (4,3,2,2,1)
  168: {1,1,1,2,4} (5,3,2,2,1)
  180: {1,1,2,2,3} (5,3,2,2,1)
		

Crossrefs

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

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

Programs

Mathematica

nn=30;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Select[Range[Prime[nn]],fdadj[primeMS[#]]==mfds[[Total[primeMS[#]]]]&]