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A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

LINKS

Table of n, a(n) for n=1..33.

EXAMPLE

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)

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[#]]]]&]

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).

Sequence in context: A018189 A239954 A296505 * A167706 A072121 A175305

Adjacent sequences:  A325280 A325281 A325282 * A325284 A325285 A325286

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Apr 17 2019

STATUS

approved

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)