|
|
A320456
|
|
Numbers whose multiset multisystem spans an initial interval of positive integers.
|
|
40
|
|
|
1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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. The n-th multiset multisystem is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the 78th multiset multisystem is {{},{1},{1,2}}.
|
|
LINKS
|
|
|
EXAMPLE
|
The sequence of terms together with their multiset multisystems begins:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
6: {{},{1}}
7: {{1,1}}
8: {{},{},{}}
9: {{1},{1}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
18: {{},{1},{1}}
19: {{1,1,1}}
21: {{1},{1,1}}
24: {{},{},{},{1}}
26: {{},{1,2}}
27: {{1},{1},{1}}
28: {{},{},{1,1}}
30: {{},{1},{2}}
32: {{},{},{},{},{}}
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100], normQ[primeMS/@primeMS[#]]&]
|
|
CROSSREFS
|
Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|