Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #17 Apr 09 2024 09:47:58
%S 3,5,7,0,37,76,48,98,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,
%T 51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,
%U 97,99,560,571,582,593,604,615,626,637,648,361,670,681,692,703,714,725,736,747,758,769,780
%N The smallest number such that the concatenation of n, a(n), n+1 is divisible by the concatenation of n and n+1.
%C For n > 10, when n starts with the digits 1, 2, 3, or 4, then a(n) = 2*n + 1. When n starts with the digits 5, 6, 7, or 8, then a(n) = 11*n + 10 for the vast majority of terms, although some outliers exist e.g., a(749) = 2251. When n starts with the digit 9, the values are somewhat more varied.
%C The maximum possible value for any term is the concatenation of n+1 and n, see the example for a(6) below. However except for a(6) and a(8), for the terms studied this only occurs four times for every order of magnitude increase in n, namely the four numbers consisting of all 9's except for the final digit of 0, 2, 6, or 8.
%C The first duplicate term is a(5) = a(18) = 37. There are 234 duplicates in the first 10000 terms.
%H Scott R. Shannon, <a href="/A371144/b371144.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) != 1 as "1"+"1"+"2" = 112 is not divisible by "1"+"2" = 12.
%e a(1) != 2 as "1"+"1"+"2" = 122 is not divisible by "1"+"2" = 12.
%e a(1) = 3 as "1"+"3"+"2" = 132 is divisible by "1"+"2" = 12.
%e a(5) = 37 as "5"+"37"+"6" = 5376 is divisible by "5"+"6" = 56.
%e a(6) = 76 as "6"+"76"+"7" = 6767 is divisible by "6"+"7" = 67. This is the first time the maximum possible value is required.
%Y Cf. A001704, A215027, A248378.
%K nonn,base
%O 1,1
%A _Scott R. Shannon_, Mar 12 2024