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.
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, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81
Offset: 1
Keywords
Examples
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.
Links
- A. Stef and G. Tenenbaum, Entiers lexicographiques, Ramanujan Journal, 2 (1998), 167-184.
Crossrefs
Except for first term 1, this is a proper subsequence of A027855.
Programs
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Mathematica
list = {1}; U = 201; For[n = 2, n <= U, n += 1, fa = FactorInteger[n]; om = Length[fa]; r = 1; ga = 1; For[j = 1, j <= om, j++, If[fa[[j, 1]] <= r, ga = 0; Break[]]; p = fa[[j, 1]]; A = fa[[j, 2]]; r = r*p^A; ]; If[ga == 1, AppendTo[list, n];]; ]; Print[list] -
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
Formula
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.
Comments