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

%I A284456 #38 Jan 24 2024 12:19:01
%S A284456 1,2,4,6,12,16,30,36,48,60,64,144,180,192,210,240,420,576,720,900,960,
%T A284456 1260,1296,1680,2310,2880,3600,4096,4620,5040,5184,6300,6480,6720,
%U A284456 12288,13860,14400,18480,20160,25200,25920,30030,32400,36864,44100,45360,46656
%N A284456 Numbers such that there is no smaller number with the same factorization shape (see Comments for details).
%C A284456 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.
%C A284456 This sequence is a subsequence of A279686 (two numbers with the same prime tower factorization class also have the same factorization shape).
%C A284456 This sequence is a subsequence of the products of primorial numbers (A025487).
%C A284456 This sequence is a supersequence of the primorial numbers (A002110).
%C A284456 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:
%C A284456               (2)        o
%C A284456                |         |
%C A284456 12 = 2^2*3 => (2) (3) => o   o
%C A284456                 \ /       \ /
%C A284456                  *         O
%C A284456 Here are the sets corresponding to some factorization shapes:
%C A284456 - Shape "1": the number 1 (this is the only finite set),
%C A284456 - Shape "2": the prime numbers (A000040),
%C A284456 - Shape "4": the prime powers of prime numbers (A053810),
%C A284456 - Shape "6": the squarefree semiprimes (A006881),
%C A284456 - Shape "16": numbers of the form p^q^r, for p,q,r primes (A217709),
%C A284456 - Shape "30": the sphenic numbers (A007304).
%C A284456 If n belongs to this sequence, then 2^n belongs to this sequence.
%C A284456 If n_1 >= ... >= n_k belong to this sequence, then Product_{i=1..k} prime(i)^n_i belongs to this sequence.
%C A284456 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).
%C A284456 All terms, except the initial term 1, are even.
%C A284456 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).
%H A284456 Rémy Sigrist, <a href="/A284456/b284456.txt">Table of n, a(n) for n = 1..10000</a>
%H A284456 Roberto Conti and Pierluigi Contucci, <a href="https://arxiv.org/abs/2204.08982">A Natural Avenue</a>, arXiv:2204.08982 [math.NT], 2022-2023.
%H A284456 Rémy Sigrist, <a href="/A284456/a284456.gp.txt">PARI program for A284456</a>
%H A284456 Rémy Sigrist, <a href="/A284456/a284456.png">Illustration of the first terms</a>
%Y A284456 Cf. A000040, A002110, A006881, A007304, A014221, A025487, A053810, A182318, A220219, A279686, A284476.
%K A284456 nonn
%O A284456 1,2
%A A284456 _Rémy Sigrist_, Mar 27 2017