A210589 Numbers which, when divided by their first digit, have their last digit as remainder.

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 210, 211, 220, 221, 230, 231, 240, 241

Offset: 1

Author

Eric Angelini (idea) and M. F. Hasler, Mar 23 2012

Keywords

• nonn
• base
• easy

Comments

Coincides with A071590 up to the 79th term, A071590(79)=310 is not in this sequence.

Charles R Greathouse IV observes that this is an automatic sequence in the terminology of Allouche & Shallit.

See A210582 for the obvious "symmetric" counterpart: first digit = x mod last digit. - M. F. Hasler, Jan 14 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..7000
• Charles R Greathouse IV, in reply to E. Angelini, Re: Divided by first digit, have last digit as remainder, SeqFan list, Mar 21 2012
• Index entries for 10-automatic sequences.

Programs

Magma

[ n: n in [1..249] | n mod d[#d] eq d[1] where d is Intseq(n) ]; // Bruno Berselli, Mar 23 2012

Mathematica

ldrQ[n_]:=Module[{idn=IntegerDigits[n],f,l},f=First[idn];l=Last[idn];Mod[n,f]==l]; Select[Range[10000],ldrQ]  (* Harvey P. Dale, Mar 21 2012 *)

PARI

is_A210589(x)=x%(x\10^(#Str(x)-1))==x%10

Python

def ok(n): s = str(n); return n > 0 and n%int(s[0]) == int(s[-1])
print([k for k in range(242) if ok(k)]) # Michael S. Branicky, Oct 20 2021