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A320634
Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).
2
1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 61, 63, 65, 69, 75, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 143, 145, 147, 151, 159, 161, 165, 169, 171, 175, 183, 185, 189, 195, 207, 223, 225, 243, 245, 247, 251, 259, 265, 267, 273
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}}.
EXAMPLE
The sequence of terms together with their multiset multisystems begins:
1: {}
3: {{1}}
7: {{1,1}}
9: {{1},{1}}
13: {{1,2}}
15: {{1},{2}}
19: {{1,1,1}}
21: {{1},{1,1}}
27: {{1},{1},{1}}
35: {{2},{1,1}}
37: {{1,1,2}}
39: {{1},{1,2}}
45: {{1},{1},{2}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
57: {{1},{1,1,1}}
61: {{1,2,2}}
63: {{1},{1},{1,1}}
65: {{2},{1,2}}
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}}
105: {{1},{2},{1,1}}
111: {{1},{1,1,2}}
113: {{1,2,3}}
117: {{1},{1},{1,2}}
131: {{1,1,1,1,1}}
133: {{1,1},{1,1,1}}
135: {{1},{1},{1},{2}}
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[1, 100, 2], normQ[primeMS/@primeMS[#]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2018
STATUS
approved