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.

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, 30, 31, 33, 36, 39, 40, 41, 42, 44, 48, 50, 51, 55, 60, 61, 62, 63, 66, 70, 71, 77, 80, 81, 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

Offset: 1

Author

M. F. Hasler, Jan 10 2016

Keywords

• nonn
• base

Comments

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.

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.

An extended version of Eric Angelini's "integears" A267085.

Sequence A263314 is a subsequence up to 120, but 121 is in A263314 and not in this sequence.

Examples

			12 is in the sequence because 1 divides 2.
213 is in the sequence because 1 divides 23.
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.)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• E. Angelini, Integears, SeqFan list, Jan. 10, 2016.

Crossrefs

Cf. A267085, A263314.

See also A080463, A080464 and A080465.

Programs

Maple

G:= proc(n) option remember;
    local t,r;
    t:= n mod 10;
    r:= procname((n-t)/10);
    [r[2],r[1]*10+t]
end proc:
G(0):= [0,0]:
filter:= proc(n)
local r;
r:= G(n);
has(r,0) or (max(r) mod min(r) = 0)
end proc:
select(filter, [$0..1000]); # Robert Israel, Jan 11 2016

Mathematica

{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 *)

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)}