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A369015 Matula-Goebel number of the prime tower factorization tree of n.

%I A369015 #18 Jan 19 2024 16:56:42
%S A369015 1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,6,3,4,3,6,2,8,2,3,4,4,
%T A369015 4,9,2,4,4,6,2,8,2,6,6,4,2,10,3,6,4,6,2,6,4,6,4,4,2,12,2,4,6,7,4,8,2,
%U A369015 6,4,8,2,9,2,4,6,6,4,8,2,10,5,4,2,12,4,4
%N A369015 Matula-Goebel number of the prime tower factorization tree of n.
%C A369015 The prime tower factorization tree of n having prime factorization n = Product p_i^e_i comprises a root vertex and beneath it child subtrees with tree numbers e_i.
%C A369015 The Matula-Goebel number represents a rooted tree (no ordering among siblings), so the primes p_i have no effect, just the exponents.
%C A369015 Runs of various consecutive equal values occur (so the same tree structure), and n = A368899(k) is the first place where a run of length >= k begins.
%H A369015 Pontus von Brömssen, <a href="/A369015/b369015.txt">Table of n, a(n) for n = 1..10000</a>
%H A369015 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A369015 a(n) = Product prime(a(e_i)) where e_i = A124010(n,i) is each exponent in the prime factorization of n.
%F A369015 Multiplicative with a(p^e) = prime(a(e)) for prime p.
%F A369015 From _Pontus von Brömssen_, Jan 15 2024: (Start)
%F A369015 a(n) = 2^k if and only if n is the product of k distinct primes.
%F A369015 a(n) = 3 if and only if n is a prime power of a prime number (A053810).
%F A369015 (End)
%e A369015 n = 274274771783272 = 2^3 * 13^(3^2) * 53^1 * 61^1 has exponents 3, 9, 1, 1 which become the following prime tower factorization tree, and corresponding Matula-Goebel number a(n) = 60:
%e A369015 .
%e A369015   n=274274771783272        a(n)=60
%e A369015       /  |   |  \        /  |   |  \
%e A369015      3   9   1   1      2   3   1   1
%e A369015      |   |              |   |
%e A369015      1   2              1   2
%e A369015          |                  |
%e A369015          1                  1
%o A369015 (PARI) a(n) = vecprod([prime(self()(e)) |e<-factor(n)[,2]]);
%Y A369015 Cf. A124010 (exponents), A369099 (first occurrences), A368899 (first runs).
%Y A369015 Cf. A053810.
%K A369015 nonn,mult,easy
%O A369015 1,2
%A A369015 _Kevin Ryde_, Jan 12 2024