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A319272
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Numbers whose prime multiplicities are distinct and whose prime indices are term of the sequence.
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1
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1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 31, 32, 37, 40, 44, 45, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 75, 76, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 103, 107, 108, 112, 121, 124, 125, 127, 128, 131, 135, 136, 144, 147, 148
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OFFSET
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1,2
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n.
Also Matula-Goebel numbers of rooted trees in which the multiplicities in the multiset of branches directly under any given node are distinct.
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LINKS
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EXAMPLE
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36 is not in the sequence because 36 = 2^2 * 3^2 does not have distinct prime multiplicities.
The sequence of terms of the sequence followed by their Matula-Goebel trees begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
7: ((oo))
8: (ooo)
9: ((o)(o))
11: ((((o))))
12: (oo(o))
16: (oooo)
17: (((oo)))
18: (o(o)(o))
19: ((ooo))
20: (oo((o)))
23: (((o)(o)))
24: (ooo(o))
25: (((o))((o)))
27: ((o)(o)(o))
28: (oo(oo))
31: (((((o)))))
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MATHEMATICA
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mgsiQ[n_]:=Or[n==1, And[UnsameQ@@Last/@FactorInteger[n], And@@Cases[FactorInteger[n], {p_, _}:>mgsiQ[PrimePi[p]]]]];
Select[Range[100], mgsiQ]
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PROG
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(PARI) is(n)={my(f=factor(n)); if(#Set(f[, 2])<#f~, 0, for(i=1, #f~, if(!is(primepi(f[i, 1])), return(0))); 1)}
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CROSSREFS
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Cf. A000081, A004111, A007097, A061775, A098859, A130091, A255231, A276625, A316793, A316794, A316795, A316796.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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