cp's OEIS Frontend

A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).

%I A279686 #27 Apr 20 2022 09:56:10
%S A279686 1,2,4,6,8,12,16,18,30,36,40,48,60,64,72,81,90,108,144,162,180,192,
%T A279686 200,210,225,240,256,280,320,324,360,405,420,432,450,500,512,540,576,
%U A279686 600,630,648,720,768,810,900,960,1260,1280,1296,1350,1400,1536,1575,1600
%N A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).
%C A279686 The prime tower factorization of a number is defined in A182318.
%C A279686 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.
%C A279686 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.
%C A279686 This sequence contains all primorial numbers (A002110).
%C A279686 This sequence contains A260548.
%C A279686 This sequence contains the terms > 0 in A014221.
%C A279686 If n appears in the sequence, then 2^n appears in the sequence.
%C A279686 If n appears in the sequence and k>=0, then A002110(k)^n appears in the sequence.
%C A279686 With the exception of term 1, this sequence contains no term from A182318.
%C A279686 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, ...
%C A279686 Here are some prime tower factorization equivalence classes:
%C A279686 - Class 1: the number one (the only finite equivalence class),
%C A279686 - Class p: the prime numbers (A000040),
%C A279686 - Class p*q: the squarefree semiprimes (A006881),
%C A279686 - Class p^p: the numbers of the form p^p with p prime (A051674),
%C A279686 - Class p^q: the numbers of the form p^q with p and q distinct primes,
%C A279686 - Class p*q*r: the sphenic numbers (A007304),
%C A279686 - Class p*q*r*s: the products of four distinct primes (A046386),
%C A279686 - Class p*q*r*s*t: the products of five distinct primes (A046387),
%C A279686 - Class p*q*r*s*t*u: the products of six distinct primes (A067885).
%H A279686 Rémy Sigrist, <a href="/A279686/b279686.txt">Table of n, a(n) for n = 1..1000</a>
%H A279686 Roberto Conti and Pierluigi Contucci, <a href="https://arxiv.org/abs/2204.08982">A Natural Avenue</a>, arXiv:2204.08982 [math.NT], 2022.
%H A279686 Rémy Sigrist, <a href="/A279686/a279686.gp.txt">PARI program for A279686</a>
%H A279686 Rémy Sigrist, <a href="/A279686/a279686.pdf">Prime tower factorization of the first terms</a>
%e A279686 2 is the least number of the form p with p prime, hence 2 appears in the sequence.
%e A279686 6 is the least number of the form p*q with p and q distinct primes, hence 6 appears in the sequence.
%e A279686 72 is the least number of the form p^q*q^p with p and q distinct primes, hence 72 appears in the sequence.
%e A279686 36000 is the least number of the form p^q*q^r*r^p with p, q and r distinct primes, hence 36000 appears in the sequence.
%Y A279686 Cf. A000040, A002110, A006881, A007304, A014221, A025487, A046386, A046387, A051674, A067885, A182318, A260548, A279690.
%K A279686 nonn
%O A279686 1,2
%A A279686 _Rémy Sigrist_, Dec 16 2016