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

%I A369099 #15 Jan 20 2024 09:35:31
%S A369099 1,2,4,6,16,12,64,30,36,48,65536,60,4096,192,144,210,
%T A369099 18446744073709551616,180,1073741824,240,576,196608,68719476736,420,
%U A369099 1296,12288,900,960,281474976710656,720
%N A369099 Index of first occurrence of n in A369015; smallest number whose prime tower factorization tree has Matula-Göbel number n.
%C A369099 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, ... .
%C A369099 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.
%C A369099 All terms are in A025487 (products of primorials).
%H A369099 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Göbel numbers</a>.
%F A369099 a(prime(n)) = 2^a(n). As a consequence, a(A007097(n)) = A014221(n).
%F A369099 a(2^n) = A002110(n).
%F A369099 a(n) = A284456(A369137(n)) = A025487(A369138(A369137(n))).
%e A369099 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.
%o A369099 (Python)
%o A369099 from sympy import factorint,nextprime,primepi
%o A369099 def A369099(n):
%o A369099     f = {A369099(primepi(p)):e for p,e in factorint(n).items()}
%o A369099     a = p = 1
%o A369099     for k in sorted(f,reverse=True):
%o A369099         for i in range(f[k]):
%o A369099             p = nextprime(p)
%o A369099             a *= p**k
%o A369099     return a
%Y A369099 Cf. A002110, A007097, A014221, A025487, A369015, A369137, A369138.
%Y A369099 A permutation of A284456.
%K A369099 nonn
%O A369099 1,2
%A A369099 _Pontus von Brömssen_, Jan 13 2024