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A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963). 2
1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202 (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 square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.
LINKS
EXAMPLE
The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
1: {}
3: {{1}}
5: {{2}}
7: {{1,1}}
11: {{3}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
26: {{},{1,2}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
39: {{1},{1,2}}
41: {{6}}
51: {{1},{4}}
53: {{1,1,1,1}}
55: {{2},{3}}
58: {{},{1,3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
69: {{1},{2,2}}
74: {{},{1,1,2}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]
CROSSREFS
Sequence in context: A111622 A356065 A082418 * A085493 A343621 A139252
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 17 2018
STATUS
approved

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Last modified February 24 05:54 EST 2024. Contains 370294 sequences. (Running on oeis4.)