A369137 -id:A369137 - cp's OEIS frontend

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

Pontus von Brömssen, Jan 13 2024

nonn

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).

			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.

Index entries for sequences related to Matula-Göbel numbers.

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

A permutation of A284456.

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

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))).

A369138 Index of A284456(n) in A025487.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 12, 13, 14, 19, 20, 21, 22, 24, 29, 33, 34, 38, 39, 43, 44, 47, 54, 57, 61, 63, 66, 67, 68, 72, 73, 74, 88, 93, 94, 101, 102, 107, 108, 114, 118, 121, 128, 129, 131, 138, 140, 142, 145, 147, 149, 161, 172, 185, 186, 187, 188, 192, 198

Offset: 1

Pontus von Brömssen, Jan 14 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A007097, A014221, A025487, A025488, A085089, A098719, A284456, A369137.

a(n) = A085089(A284456(n)).
A025487(a(n)) = A284456(n).
a(A369137(A007097(n))) = A025488(A014221(n-1)).
a(A369137(2^n)) = A098719(n+1).

A369136 Matula-Göbel number of the prime tower factorization tree of A284456(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 15, 18, 14, 16, 20, 24, 21, 30, 27, 28, 36, 25, 40, 32, 42, 45, 13, 48, 60, 35, 54, 50, 56, 26, 72, 63, 80, 84, 90, 70, 64, 75, 39, 81, 100, 49, 120, 96, 52, 11, 108, 112, 126, 105, 135, 144, 140, 78, 22, 168, 150, 98, 160

Offset: 1