A347541 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.

11, 12, 15, 24, 36, 1352, 1734, 143143, 167334, 16673334, 1666733334, 166667333334, 16666673333334, 1666666733333334, 142857143142857143, 166666667333333334, 16666666673333333334, 1666666666733333333334, 166666666667333333333334, 16666666666673333333333334, 1666666666666733333333333334, 142857142857143142857142857143

Offset: 1

Author

Bernard Schott, Oct 11 2021

Keywords

• nonn
• base

Comments

Problem proposed on French site Diophante (see link).

We have to solve Diophantine equation q.r = q*10^m + r = k * q * r where m = length(q) = length(r). Some results:

k can only take values 2, 3, 6, 7, 11, and ratio r/q = 1, 2, 4 or 5.

There are exactly 3 subsequences of terms that are solutions, one finite and two infinites:

-> Finite subsequence: 11, 12, 15, 36, 1352.

-> 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, ... }.

-> 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).

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.

Note that the first five terms are the 2-digit Zuckerman numbers (A007602).

Examples

			One example for each possible value of k = q.r / (q*r).
a(1) = 11 and 11/(1*1) = 11.
a(2) = 12 and 12/(1*2) = 6.
a(5) = 36 and 36/(3*6) = 2.
a(7) = 1734 and 1734/(17*34) = 3.
a(8) = 143143 and 143143/(143*143) = 7.
		

Links

• Diophante, A1945 - Concaténations en tous genres (in French).
• Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Crossrefs

Cf. A007602, A147553, A348589.