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A284456 Numbers such that there is no smaller number with the same factorization shape (see Comments for details). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 "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

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.

Sequence in context: A266543 A330711 A220219 * A233968 A120453 A326438

Adjacent sequences:  A284453 A284454 A284455 * A284457 A284458 A284459

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Mar 27 2017

STATUS

approved

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Last modified July 8 01:48 EDT 2020. Contains 335502 sequences. (Running on oeis4.)