OFFSET
1,2
COMMENTS
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.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The terms together with their prime factorizations and unique factorizations into terms of A275700 begin:
1 = = ()
2 = 2 = (2)
4 = 2*2 = (2*2)
6 = 2*3 = (6)
8 = 2*2*2 = (2*2*2)
10 = 2*5 = (10)
12 = 2*2*3 = (2*6)
16 = 2*2*2*2 = (2*2*2*2)
20 = 2*2*5 = (2*10)
22 = 2*11 = (22)
24 = 2*2*2*3 = (2*2*6)
32 = 2*2*2*2*2 = (2*2*2*2*2)
34 = 2*17 = (34)
36 = 2*2*3*3 = (6*6)
40 = 2*2*2*5 = (2*2*10)
42 = 2*3*7 = (42)
44 = 2*2*11 = (2*22)
48 = 2*2*2*2*3 = (2*2*2*6)
60 = 2*2*3*5 = (6*10)
62 = 2*31 = (62)
64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
68 = 2*2*17 = (2*34)
72 = 2*2*2*3*3 = (2*6*6)
80 = 2*2*2*2*5 = (2*2*2*10)
82 = 2*41 = (82)
84 = 2*2*3*7 = (2*42)
88 = 2*2*2*11 = (2*2*22)
96 = 2*2*2*2*2*3 = (2*2*2*2*6)
MATHEMATICA
nn=100;
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
s=Table[Times@@Prime/@Divisors[n], {n, nn}];
m=Max@@Table[Select[Range[2, k], prix[#] == Divisors[Last[prix[#]]]&], {k, nn}];
Join@@Position[Table[Length[Select[facs[n], SubsetQ[s, Union[#]]&]], {n, m}], 1]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 22 2024
STATUS
approved