%I #48 Jan 18 2022 05:49:44
%S 11,12,15,24,36,1352,1734,143143,167334,16673334,1666733334,
%T 166667333334,16666673333334,1666666733333334,142857143142857143,
%U 166666667333333334,16666666673333333334,1666666666733333333334,166666666667333333333334,16666666666673333333333334,1666666666666733333333333334,142857142857143142857142857143
%N Numbers q.r such that q*r divides q.r, when q and r have the same number of digits, "." means concatenation, and r may not begin with 0.
%C Problem proposed on French site Diophante (see link).
%C We have to solve Diophantine equation q.r = q*10^m + r = k * q * r where m = length(q) = length(r). Some results:
%C k can only take values 2, 3, 6, 7, 11, and ratio r/q = 1, 2, 4 or 5.
%C There are exactly 3 subsequences of terms that are solutions, one finite and two infinites:
%C -> Finite subsequence: 11, 12, 15, 36, 1352.
%C -> Infinite subsequence with k = 7 and r = q = (10^(6h-3)+1)/7, h>=1 (A147553 \ {1}), hence terms are (10^(6h-3)+1)^2/7 for h>=1: {143143, 142857143142857143, ... }.
%C -> Infinite subsequence with k = 3 and r = 2q, with q = (10^h+2)/6, r = (10^h+2)/3 for h>= 1: {24, 1734, 167334, 16673334, ...} (A348589).
%C Consequence: integers q.q that are divisible by q*q are exactly integers such that q is a term of A147553. If q = A147553(1) = 1, then 11/(1*1) = 11, while for q = A147553(n), n>=2, then q.q / (q*q) = 7.
%C Note that the first five terms are the 2-digit Zuckerman numbers (A007602).
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/1024-a1945-concatenations-en-tous-genres">A1945 - Concaténations en tous genres</a> (in French).
%H Giovanni Resta, <a href="https://www.numbersaplenty.com/set/Zuckerman_number/">Zuckerman numbers</a>, Numbers Aplenty.
%e One example for each possible value of k = q.r / (q*r).
%e a(1) = 11 and 11/(1*1) = 11.
%e a(2) = 12 and 12/(1*2) = 6.
%e a(5) = 36 and 36/(3*6) = 2.
%e a(7) = 1734 and 1734/(17*34) = 3.
%e a(8) = 143143 and 143143/(143*143) = 7.
%Y Cf. A007602, A147553, A348589.
%K nonn,base
%O 1,1
%A _Bernard Schott_, Oct 11 2021