A346447 - OEIS

%I #56 Sep 14 2021 04:29:20

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,30,31,32,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,

%U 53,54,55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73

%N Positive integers k such that the positive divisors of k <= sqrt(k) form a linear recurrence of order at most 2.

%C A complete classification of all elements of this list has been obtained. In particular, all elements of this list have at most 3 distinct prime divisors.

%C Consequently, the above list has asymptotic density zero in the natural numbers.

%C Any prime belongs to this sequence since the tuple (1) vacuously satisfies some second-order linear recurrence.

%H A. Anas Chentouf, <a href="https://arxiv.org/abs/2108.13173">Linear Recurrences of Order at Most Two in Small Divisors</a>, arXiv:2108.13173 [math.NT], 2021.

%e 36, whose divisors are (1,2,3,4,6,9,12,28,36), is the smallest non-recurrent number as (1,2,3,4,6) do not satisfy a second-order linear recurrence.

%e 60 is in the sequence as its divisors <= sqrt(60) are (1, 2, 3, 4, 5, 6). These numbers satisfy b(n) = 2*b(n-1) - b(n-2), b(1) = 1, b(2) = 2. - _David A. Corneth_, Aug 14 2021

%o (PARI) is(n) = { my(f = factor(n), d, m, linrec, nd); if(#f~ > 3, return(0)); if(numdiv(f) <= 6, return(1) ); d = divisors(n); d = vector((#d + 1)\2, i, d[i]); m = [d[2],d[1];d[3],d[2]]; if(matdet(m) != 0, linrec = matsolve([d[2],d[1];d[3],d[2]], [d[3];d[4]]); nd = vector(#d); nd[1] = d[1]; nd[2] = d[2]; for(i = 3, #nd, nd[i] = nd[i-1] * linrec[1,1] + nd[i-2]*linrec[2,1]; ); return(nd==d) , my(r = d[2]/d[1]); for(i = 3, #d, if(d[i-1]*r != d[i], return(0)); ); return(1) ) } \\ _David A. Corneth_, Aug 14 2021

%Y Cf. A038548.

%K nonn

%O 1,2

%A _A. Anas Chentouf_, Aug 14 2021