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Numbers m such that the result of prepending a zero digit to m, removing the least significant digit D, and prepending D, is divisible by m.
1

%I #30 Aug 11 2024 14:41:34

%S 1,2,3,4,5,6,7,8,9,27,37,101,202,303,404,505,606,707,808,909,1084,

%T 1355,1626,1897,2168,2439,10101,10582,10989,11583,11655,12987,13986,

%U 15444,15873,16317,18648,19305,20202,20979,21164,23166,25641,26455,27027,30303,30888

%N Numbers m such that the result of prepending a zero digit to m, removing the least significant digit D, and prepending D, is divisible by m.

%C For palindromic numbers the ratio is equal to 10.

%H Paolo P. Lava and Giovanni Resta, <a href="/A256005/b256005.txt">Table of n, a(n) for n = 1..504</a> (terms < 10^34, first 100 terms from Paolo P. Lava)

%H P. De Geest, <a href="https://www.worldofnumbers.com/em174.htm">Hopping Numerals</a>

%e 37 is in the sequence because prepending a 0 gives 037, removing the least significant digit 7 then gives 03, and finally prepending the 7 gives 703, which is divisible by 37.

%e 25641 is in the sequence because prepending a 0 gives 025641, removing the least significant digit 1 then gives 025641, and finally prepending the 1 gives 102564, which is divisible by 25641.

%p P:=proc(q) local a,n; for n from 1 to q do

%p a:=(n mod 10)*10^(ilog10(n)+1)+trunc(n/10);

%p if not a=n then if type(a/n,integer) then print(n);

%p fi; fi; od; end: P(10^7);

%t Select[Range@31000,IntegerQ[FromDigits[RotateRight[Insert[IntegerDigits[#],0,1]]]/#]&] (* _Ivan N. Ianakiev_, May 28 2015 *)

%o (PARI) is(n)=my(k=n%10*10^#digits(n)+n\10); k>n && k%n==0 \\ _Charles R Greathouse IV_, May 08 2015

%Y Cf. A034089.

%K nonn,base

%O 1,2

%A _Paolo P. Lava_, May 06 2015

%E 'Name' and 'Examples' sections reworded by _Ivan N. Ianakiev_, Aug 05 2015 (following the suggestion of _Jon E. Schoenfield_)