A369099 Index of first occurrence of n in A369015; smallest number whose prime tower factorization tree has Matula-Göbel number n.

1, 2, 4, 6, 16, 12, 64, 30, 36, 48, 65536, 60, 4096, 192, 144, 210, 18446744073709551616, 180, 1073741824, 240, 576, 196608, 68719476736, 420, 1296, 12288, 900, 960, 281474976710656, 720

Offset

1,2

Comments

After a(30), the sequence continues 2^65536, 2*3*5*7*11, 2^16*3^2, 2^64*3, 2^6*3^4, 2^2*3^2*5*7, 2^60, 2^30*3, 2^12*9, 2^4*3*5*7, 2^4096, ... .

a(n) can be determined recursively as follows. Let n = Product_{i>=1} p_i^e_i, where p_i is the i-th prime. Take f_1 >= f_2 >= ... >= f_k so that the number a(i) occurs e_i times for i >= 1. Then a(n) = Product_{i>=1} p_i^f_i.

All terms are in A025487 (products of primorials).

Links

Table of n, a(n) for n=1..30.

Index entries for sequences related to Matula-Göbel numbers.

Formula

a(prime(n)) = 2^a(n). As a consequence, a(A007097(n)) = A014221(n).

a(2^n) = A002110(n).

a(n) = A284456(A369137(n)) = A025487(A369138(A369137(n))).

Example

Using the method described in the comments for n = 20 = p(1)^2*p(3)^1, the exponents f_i shall include the term a(1)=1 twice and the term a(3)=4 once, i.e., (f_1, f_2, f_3) = (4, 1, 1), so a(20) = p(1)^4*p(2)^1*p(3) = 240.

Prog

(Python)
from sympy import factorint, nextprime, primepi
def A369099(n):
    f = {A369099(primepi(p)):e for p, e in factorint(n).items()}
    a = p = 1
    for k in sorted(f, reverse=True):
        for i in range(f[k]):
            p = nextprime(p)
            a *= p**k
    return a

Crossrefs

Cf. A002110, A007097, A014221, A025487, A369015, A369137, A369138.

A permutation of A284456.

Sequence in context: A099315 A005179 A037019 * A341668 A326782 A358126

Adjacent sequences: A369096 A369097 A369098 * A369100 A369101 A369102

Keyword

nonn

Author

Pontus von Brömssen, Jan 13 2024

Status

approved