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).
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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Apr 17 2019
STATUS
approved