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A284456
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Numbers such that there is no smaller number with the same factorization shape (see Comments for details).
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7
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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
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OFFSET
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1,2
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COMMENTS
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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
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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).
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LINKS
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Roberto Conti and Pierluigi Contucci, A Natural Avenue, arXiv:2204.08982 [math.NT], 2022-2023.
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CROSSREFS
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Cf. A000040, A002110, A006881, A007304, A014221, A025487, A053810, A182318, A220219, A279686, A284476.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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