A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

Offset: 1

Author

Gus Wiseman, Apr 23 2018

Keywords

• nonn

Comments

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

Examples

			Sequence of aperiodic rooted trees begins:
01 o
02 (o)
03 ((o))
05 (((o)))
06 (o(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
18 (o(o)(o))
20 (oo((o)))
22 (o(((o))))
24 (ooo(o))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))
		

Crossrefs

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

Programs

Mathematica

zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
Select[Range[100],zapQ]