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

Author

Pontus von Brömssen, Jan 13 2024

Keywords

• nonn

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).

Examples

			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.
		

Links

• Index entries for sequences related to Matula-Göbel numbers.

Crossrefs

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

A permutation of A284456.

Programs

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

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))).