login

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”).

Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.
3

%I #16 Apr 20 2021 10:27:22

%S 10,15,18,20,30,40,45,50,60,70,80,90,100,111,120,126,150,180,200,222,

%T 240,250,285,300,333,360,400,444,450,480,500,555,600,666,700,750,777,

%U 800,888,900,999,1000,1041,1110,1185,1200,1260,1395,1443,1500,1554,1665

%N Numbers k > 9 such that the number m formed by inserting a digit 0 between each pair of digits in k is divisible by k.

%C One-digit terms are not considered since no 0 digits can be inserted.

%C If k is a term then so is k*10^i, i > 0.

%C If k is a term then so is k*i, 2 <= i <= 9 as long as no carry occurs in the multiplication.

%C The number of terms with n digits is (12, 29, 51, 107, 149, 240, 308, 438, 566, 789, 1007), 2 <= n <= 12.

%e 18 is a term because 108/18=6, and so is 1185 because 1010805/1185=853.

%e 10101/111=91, 1010100/1110=910, 101010000/11100=9100, ... so 111, 1110, 11100, ... are all terms.

%e 1000401/1041=961 and 2000802/2082=961 so 1041 and 2082 are terms but 3123 is not since it does not divide 3010203.

%Y Cf. A051022, A285176, A343551, A343552.

%Y Cf. A062846 (binary), A062891 (ternary).

%K nonn,base

%O 1,1

%A _Lars Blomberg_, Apr 19 2021