cp's OEIS Frontend

More informally, the prime tower factorization of a recursive divisor of n can be obtained by removing branches from the prime tower factorization of n (the prime tower factorization of a number is defined in A182318).

A recursive divisor of n is also a divisor of n, hence a(n)<=A000005(n) for any n, with equality iff n is cubefree (i.e. n belongs to A004709).

A recursive divisor of n is also a (1+e)-divisor of n, hence a(n)<=A049599(n) for any n, with equality iff the p-adic valuation of n is cubefree for any prime p dividing n.

This sequence first differs from A049599 at n=256: a(256)=4 whereas A049599(256)=5; note that 256=2^(2^3), and 2^3 is not cubefree.

To compute a(n): multiply (with multiplicity) each prime number appearing in the prime tower factorization of n (see A182318 for definition).

a(n) = n if n is squarefree.

a(n) <= A000026(n) for any n>0.

First differs from A000026 at n=256: a(256)=12 and A000026(256)=16.

If n = p_1 * p_2 * ... * p_k with p_1, p_2, ..., p_k primes, then a(p_1 ^ p_2 ^ ... ^ p_k) = n.

We say that two numbers, say X and Y, have the same factorization shape iff X and Y have the same number of distinct prime factors, say x_1, ..., x_k and y_1, ..., y_k, and there is a permutation f on {1,..,k} such that, for any i between 1 and k, the x_i-adic valuation of X has the same factorization shape as the y_f(i)-adic valuation of Y.

This sequence is a subsequence of A279686 (two numbers with the same prime tower factorization class also have the same factorization shape).

This sequence is a subsequence of the products of primorial numbers (A025487).

This sequence is a supersequence of the primorial numbers (A002110).

The factorization shape of n can be identified with the rooted tree underlying the prime tower factorization of n (see A182318 for the definition of prime tower factorization); for example:

(2) o

| |

12 = 2^2*3 => (2) (3) => o o

\ / \ /

* O

Here are the sets corresponding to some factorization shapes:

- Shape "1": the number 1 (this is the only finite set),

- Shape "2": the prime numbers (A000040),

- Shape "4": the prime powers of prime numbers (A053810),

- Shape "6": the squarefree semiprimes (A006881),

- Shape "16": numbers of the form p^q^r, for p,q,r primes (A217709),

- Shape "30": the sphenic numbers (A007304).

If n belongs to this sequence, then 2^n belongs to this sequence.

If n_1 >= ... >= n_k belong to this sequence, then Product_{i=1..k} prime(i)^n_i belongs to this sequence.

This sequence is not a subsequence of A220219 (48 belongs to this sequence, hence 2^48 belongs to this sequence; but 48+1 is not prime, so 2^48 does not belong to A220219; in fact, a(9)=48 is the first term of the sequence not one less than a prime, and a(681)=2^48 is the first term of this sequence not in A220219).

All terms, except the initial term 1, are even.

If a(n) <= 2^a(m), then the p-adic valuation of a(n) is <= a(m) for any prime p; this property implies that, provided you know the first m terms, you can generate all terms up to 2^a(m) by enumerating the products of primorials <= 2^a(m) with exponents in {a(1), ..., a(m)}; hence, starting with the initial term a(1)=1, after n iterations, you have all terms <= A014221(n).

The prime tower factorization of a number is defined in A182318.

We say that two numbers, say n and m, belong to the same prime tower factorization equivalence class iff there is a permutation of the prime numbers, say f, such that replacing each prime p by f(p) in the prime tower factorization of n leads to m.

The notion of prime tower factorization equivalence class can be seen as a generalization of the notion of prime signature; thereby, this sequence can be seen as an equivalent of A025487.

This sequence contains all primorial numbers (A002110).

This sequence contains A260548.

This sequence contains the terms > 0 in A014221.

If n appears in the sequence, then 2^n appears in the sequence.

If n appears in the sequence and k>=0, then A002110(k)^n appears in the sequence.

With the exception of term 1, this sequence contains no term from A182318.

Odd numbers appearing in this sequence: 1, 81, 225, 405, 1575, 2025, 2835, 6125, 10125, 11025, 14175, 15625, 16875, 17325, 31185, 33075, 50625, 67375, 70875, 99225, ...

Here are some prime tower factorization equivalence classes:

- Class 1: the number one (the only finite equivalence class),

- Class p: the prime numbers (A000040),

- Class p*q: the squarefree semiprimes (A006881),

- Class p^p: the numbers of the form p^p with p prime (A051674),

- Class p^q: the numbers of the form p^q with p and q distinct primes,

- Class p*q*r: the sphenic numbers (A007304),

- Class p*q*r*s: the products of four distinct primes (A046386),

- Class p*q*r*s*t: the products of five distinct primes (A046387),

- Class p*q*r*s*t*u: the products of six distinct primes (A067885).

See A182318 for the definition of the prime tower factorization of a number.

a(n) >= A006530(n) for any n>0.

The prime tower factorization of a number is defined in A182318.

This sequence is a self-inverse multiplicative permutation of the natural numbers.

This sequence has infinitely many fixed points (A300957); for any k > 0, at least one of k or 2^k * 3^a(k) is a fixed point.

This sequence is a recursive version of A182318.

This sequence has connections with A300948.

This sequence establishes a bijection between the natural numbers and A182318.

Records: 1, 3, 5, 27, 243, 7625597484987, 38127987424935, 53379182394909, 7450580596923828125, 22351741790771484375, ..., .

Records appear at: 1, 2, 3, 4, 8, 16, 48, 80, 81, 162, 256, 768, 1280, 1792, 2304, 6400, 6561, 13122, 26244, ..., . Robert G. Wilson v, Dec 14 2016

See A182318 for the definition of the prime tower factorization of a number.

a(n) <= A020639(n) for any n>0.

Two consecutive numbers, say n and n+1, cannot share a prime factor (gcd(n, n+1)=1). However, their prime tower factorizations can share some prime numbers; this is the case iff a(n)>1 (see A182318 for the definition of the prime tower factorization of a number).

If p is prime, then a(p-1) = a(p) = 1.

If p is an odd prime, then a(p^2) = 2.

This sequence contains a multiple of p for any prime p:

- let m = A074792(p)^p-1,

- m is a multiple of p, hence p divides A279513(m),

- m+1 = A074792(p)^p, hence p divides A279513(m+1),

- hence p divides gcd(A279513(m), A279513(m+1)) = a(m).

This sequence contains infinitely many distinct values; see A284821 for these distinct values in order of appearance, and A284822 for the corresponding indexes.

We use the definition of recursive divisor given in A282446.

More informally, the prime tower factorization of T(n, k) is the intersection of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).

This sequence has connections with the classical GCD (A003989).

For any i > 0, j > 0 and k > 0:

- T(i, j) = 1 iff gcd(i, j) = 1,

- A007947(T(i, j)) = A007947(gcd(i, j)),

- T(i, j) >= 1,

- T(i, j) <= min(i, j),

- T(i, j) <= gcd(i, j),

- T(i, 1) = 1,

- T(i, i) = i,

- T(i, j) = T(j, i) (the sequence is commutative),

- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),

- T(i, i*j) <= i,

- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),

- T(i, 2*i) = A259445(i).

See also A287958 for the LCM equivalent.