A284456 Numbers such that there is no smaller number with the same factorization shape (see Comments for details).

1, 2, 4, 6, 12, 16, 30, 36, 48, 60, 64, 144, 180, 192, 210, 240, 420, 576, 720, 900, 960, 1260, 1296, 1680, 2310, 2880, 3600, 4096, 4620, 5040, 5184, 6300, 6480, 6720, 12288, 13860, 14400, 18480, 20160, 25200, 25920, 30030, 32400, 36864, 44100, 45360, 46656

Offset: 1

Author

Rémy Sigrist, Mar 27 2017

Keywords

• nonn

Comments

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).

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Roberto Conti and Pierluigi Contucci, A Natural Avenue, arXiv:2204.08982 [math.NT], 2022-2023.
• Rémy Sigrist, PARI program for A284456
• Rémy Sigrist, Illustration of the first terms

Crossrefs

Cf. A000040, A002110, A006881, A007304, A014221, A025487, A053810, A182318, A220219, A279686, A284476.