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

1,2

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.

Links

Table of n, a(n) for n=1..64.

Example

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))))

Mathematica

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

Crossrefs

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

Sequence in context: A162309 A014593 A261088 * A317709 A034044 A047447

Adjacent sequences: A303428 A303429 A303430 * A303432 A303433 A303434

Keyword

nonn

Author

Gus Wiseman, Apr 23 2018

Status

approved