OFFSET
1,1
COMMENTS
A fixed point of a sequence y is an index y(i) = i. A fixed point of a partition is unique if it exists.
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.
EXAMPLE
The terms together with their prime indices begin:
4: (1,1)
8: (1,1,1)
16: (1,1,1,1)
27: (2,2,2)
32: (1,1,1,1,1)
45: (3,2,2)
54: (2,2,2,1)
63: (4,2,2)
64: (1,1,1,1,1,1)
81: (2,2,2,2)
90: (3,2,2,1)
99: (5,2,2)
108: (2,2,2,1,1)
117: (6,2,2)
126: (4,2,2,1)
128: (1,1,1,1,1,1,1)
For example, the partition (3,2,2,1) with Heinz number 90 has a fixed point at the second position, but its conjugate (4,3,1) has no fixed points, so 90 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Select[Range[100], pq[Reverse[primeMS[#]]]>0&& pq[conj[Reverse[primeMS[#]]]]==0&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
STATUS
approved