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

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, 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, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Author

Kevin Ryde, Jan 12 2024

Keywords

• nonn
• mult
• easy

Comments

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.

The Matula-Goebel number represents a rooted tree (no ordering among siblings), so the primes p_i have no effect, just the exponents.

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.

Examples

			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:
.
  n=274274771783272        a(n)=60
      /  |   |  \        /  |   |  \
     3   9   1   1      2   3   1   1
     |   |              |   |
     1   2              1   2
         |                  |
         1                  1
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A124010 (exponents), A369099 (first occurrences), A368899 (first runs).

Cf. A053810.

Programs

PARI

a(n) = vecprod([prime(self()(e)) |e<-factor(n)[,2]]);

Formula

a(n) = Product prime(a(e_i)) where e_i = A124010(n,i) is each exponent in the prime factorization of n.

Multiplicative with a(p^e) = prime(a(e)) for prime p.

From Pontus von Brömssen, Jan 15 2024: (Start)

a(n) = 2^k if and only if n is the product of k distinct primes.

a(n) = 3 if and only if n is a prime power of a prime number (A053810).

(End)