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A319837 Numbers whose distinct prime indices are pairwise indivisible and whose own prime indices span an initial interval of positive integers. 6
1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 35, 37, 45, 49, 53, 61, 64, 69, 75, 81, 89, 91, 95, 113, 128, 131, 135, 141, 143, 145, 151, 161, 165, 169, 175, 207, 223, 225, 243, 245, 247, 251, 256, 265, 281, 299, 309, 311, 329, 343, 355, 359, 361, 375, 377, 385 (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 multiset multisystem with MM-number n 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 multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of not necessarily strict antichains of multisets spanning an initial interval of positive integers.
LINKS
EXAMPLE
The sequence of multisystems whose MM-numbers belong to the sequence begins:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
7: {{1,1}}
8: {{},{},{}}
9: {{1},{1}}
13: {{1,2}}
15: {{1},{2}}
16: {{},{},{},{}}
19: {{1,1,1}}
27: {{1},{1},{1}}
32: {{},{},{},{},{}}
35: {{2},{1,1}}
37: {{1,1,2}}
45: {{1},{1},{2}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
61: {{1,2,2}}
64: {{},{},{},{},{},{}}
69: {{1},{2,2}}
75: {{1},{2},{2}}
81: {{1},{1},{1},{1}}
89: {{1,1,1,2}}
91: {{1,1},{1,2}}
95: {{2},{1,1,1}}
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]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[200], And[normQ[primeMS/@primeMS[#]], stableQ[primeMS[#], Divisible]]&]
CROSSREFS
Sequence in context: A032876 A032843 A023774 * A321699 A349759 A050271
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 16 2018
STATUS
approved

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Last modified June 24 23:28 EDT 2024. Contains 373691 sequences. (Running on oeis4.)