cp's OEIS Frontend

A267086 Numbers such that the number formed by digits in even positions divides, or is divisible by, the number formed by the digits in odd positions; zero allowed.

%I A267086 #27 Dec 23 2024 14:53:44
%S A267086 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,24,26,28,
%T A267086 30,31,33,36,39,40,41,42,44,48,50,51,55,60,61,62,63,66,70,71,77,80,81,
%U A267086 82,84,88,90,91,93,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,122,124,126,128,132,135
%N A267086 Numbers such that the number formed by digits in even positions divides, or is divisible by, the number formed by the digits in odd positions; zero allowed.
%C A267086 The initial 0 is included by convention. The single-digit numbers are included with the reasoning that the number formed by digits in even positions is zero, and thus divisible by (= a multiple of) any other number, and here in particular the number formed by first digit.
%C A267086 By "digits in odd positions" we mean the first (most significant), third, fifth, etc. digits; e.g., for the numbers 12345 or 123456 this would be 135.
%C A267086 An extended version of _Eric Angelini_'s "integears" A267085.
%C A267086 Sequence A263314 is a subsequence up to 120, but 121 is in A263314 and not in this sequence.
%H A267086 Robert Israel, <a href="/A267086/b267086.txt">Table of n, a(n) for n = 1..10000</a>
%H A267086 E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-January/015951.html">Integears</a>, SeqFan list, Jan. 10, 2016.
%e A267086 12 is in the sequence because 1 divides 2.
%e A267086 213 is in the sequence because 1 divides 23.
%e A267086 1020 is in the sequence because 12 divides 00 = 0. (Any number divides 0 therefore any number which has every other digit equal to zero is in the sequence.)
%p A267086 G:= proc(n) option remember;
%p A267086     local t,r;
%p A267086     t:= n mod 10;
%p A267086     r:= procname((n-t)/10);
%p A267086     [r[2],r[1]*10+t]
%p A267086 end proc:
%p A267086 G(0):= [0,0]:
%p A267086 filter:= proc(n)
%p A267086 local r;
%p A267086 r:= G(n);
%p A267086 has(r,0) or (max(r) mod min(r) = 0)
%p A267086 end proc:
%p A267086 select(filter, [$0..1000]); # _Robert Israel_, Jan 11 2016
%t A267086 {0}~Join~Select[Range@ 135, Total@ Boole@ Map[ReplaceAll[List -> Divisible], {#, Reverse@ #} /. {_, 0} -> Nothing] &@ Map[FromDigits@ Reverse@ # &, {Map[First, #], Map[Last, #]}] &@ Which[Length@ # < 2, {#}, EvenQ@ Length@ #, Partition[#, 2, 2], True, Append[Partition[#, 2, 2], {Last@ #, 0}]] &@ Reverse@ IntegerDigits@ # > 0 &] (* _Michael De Vlieger_, Jan 11 2016 *)
%o A267086 (PARI) is(n,d=digits(n))={if(n=d*matrix(#d,2,z,s,if(z==Mod(s,2),10^((#d-z)\2))), n[2] && n[1]%n[2]==0 || n[2]%n[1]==0, 1)}
%Y A267086 Cf. A267085, A263314.
%Y A267086 See also A080463, A080464 and A080465.
%K A267086 nonn,base
%O A267086 1,3
%A A267086 _M. F. Hasler_, Jan 10 2016