A393587 Matula-Goebel numbers of two-leaf rooted trees.

4, 6, 7, 9, 10, 13, 15, 17, 22, 23, 25, 29, 33, 41, 47, 55, 59, 62, 79, 83, 93, 97, 109, 121, 137, 155, 179, 211, 254, 257, 277, 293, 341, 381, 401, 431, 487, 509, 599, 635, 661, 773, 907, 961, 1063, 1297, 1397, 1418, 1609, 1621, 1787, 1913, 2127, 2293, 2621, 2749, 3001, 3469, 3545, 3637, 3937

Offset

1,1

Comments

Each positive integer n corresponds to a unique rooted tree via the Matula-Goebel correspondence.

If an integer k belongs to the sequence, the prime number with index k is also a member of the sequence.

Any element of the sequence is obtained by applying a finite iteration of prime(-) to the product of two elements of A007097 together: the sequence is the three-dimensional array {prime^k(A007097(b)*A007097(c)), k>=0, b>=1, c>=1} arranged in increasing order.

Links

Michel Marcus, Table of n, a(n) for n = 1..311

Zizhen Chen, Matula number generator.

Sean A. Irvine, Java program (github)

Dominique Manchon, Une remarque sur l'arborification de Matula, arXiv:2602.03150 [math.NT], 2026, see p. 6.

Dominique Manchon, Une remarque sur l'arborification de Matula, hal-05493935, 2026, see p. 6.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. vol. 10, 1968, p. 273.

Example

a(1) = 4 = 2*2, the Matula-Goebel number of the V-shaped tree [..] with three vertices and one root.
a(2) = 6 = 2*3 = 2*prime(2), the Matula-Goebel number of the four-vertex rooted tree[[.].] obtained by grafting the forest [.]. on a common root.
a(3) = 7 = prime(4), the Matula-Goebel number of the Y-shaped tree [[..]].
a(17) = 59 = prime^3(4), the Matula-Goebel number of the tall Y-shaped tree [[[[..]]]].

Prog

(PARI) lista(nn) = my(list1 = List(), k=1); while(k<nn, k = prime(k); if (k<nn, listput(list1, k))); my(s = select(x->(x<nn), setbinop((x, y)->x*y, Vec(list1))), list = List()); while(#s, for (i=1, #s, listput(list, s[i])); s = select(x->(x<nn), apply(prime, s)); ); Vec(vecsort(list)); \\ Michel Marcus, Mar 13 2026

Crossrefs

Cf. A007097 (Matula-Goebel numbers of one-leaf rooted trees).

Sequence in context: A246362 A069909 A189715 * A101993 A370267 A002481

Adjacent sequences: A393584 A393585 A393586 * A393588 A393589 A393590

Keyword

nonn

Author

Dominique Manchon, Feb 22 2026

Status

approved