cp's OEIS Frontend

A361232 Numbers m such that the increasing sequence of divisors of m, regarded as words on the finite alphabet of its prime factors, is ordered lexicographically.

%I A361232 #32 Mar 30 2023 09:15:56
%S A361232 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T A361232 29,31,32,33,34,35,37,38,39,41,42,43,44,46,47,49,50,51,52,53,54,55,57,
%U A361232 58,59,61,62,64,65,66,67,68,69,71,73,74,75,76,77,78,79,81
%N A361232 Numbers m such that the increasing sequence of divisors of m, regarded as words on the finite alphabet of its prime factors, is ordered lexicographically.
%C A361232 Let p_1^e_1 * p_2^e_2 * ... * p_r^e_r be the prime factorization of a number, with primes p_1 < p_2 < ... < p_r and e_j > 0. Then the number is in this sequence if and only if p_i > Product_{j < i} p_j^e_j for i=1..r.
%C A361232 Even though most small numbers belong to this sequence, this sequence has zero density (see the formula below).
%C A361232 Except for the first term 1, this sequence is a proper subsequence of A027855, where the condition on the prime factors only applies to the largest prime factor.
%C A361232 Stef and Tenenbaum call these numbers lexicographical.
%H A361232 A. Stef and G. Tenenbaum, <a href="https://tenenb.perso.math.cnrs.fr/PPP/NbsLexicographiques.pdf">Entiers lexicographiques</a>, Ramanujan Journal, 2 (1998), 167-184.
%F A361232 Stef and Tenenbaum show that the count of these integers up to x is asymptotic to c*x/(log x)^b, where b is approximately 0.2228 and c is a positive constant. As a result, the n-th term of this sequence is asymptotic to (1/c)*n*(log n)^b.
%e A361232 The number 12 is not in this sequence: Writing the divisors of 12 as 3^a*2^b where a=0,1 and b=0,1,2, the lexicographical order corresponds to (a,b)=(0,0),(0,1),(0,2),(1,0),(1,1),(1,2), i.e., 1,2,4,3,6,12, which is not the natural order.
%t A361232 list = {1}; U = 201;
%t A361232 For[n = 2, n <= U, n += 1, fa = FactorInteger[n]; om = Length[fa];
%t A361232   r = 1; ga = 1;
%t A361232   For[j = 1, j <= om, j++,
%t A361232    If[fa[[j, 1]] <= r, ga = 0; Break[]];
%t A361232    p = fa[[j, 1]]; A = fa[[j, 2]];
%t A361232    r = r*p^A;
%t A361232    ];
%t A361232   If[ga == 1, AppendTo[list, n];];
%t A361232   ];
%t A361232 Print[list]
%o A361232 (PARI) isok(n)={my(f=factor(n), h=#f~, s=0); forvec(X=vector(h, i, [0,f[h+1-i,2]]), my(p=prod(i=1, h, f[h+1-i,1]^X[i])); if(p<s, return(0)); s=p); 1} \\ _Andrew Howroyd_, Mar 05 2023
%Y A361232 Except for first term 1, this is a proper subsequence of A027855.
%K A361232 nonn
%O A361232 1,2
%A A361232 _Andreas Weingartner_, Mar 05 2023