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A267086
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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.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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.
Sequence A263314 is a subsequence up to 120, but 121 is in A263314 and not in this sequence.
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LINKS
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E. Angelini, Integears, SeqFan list, Jan. 10, 2016.
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EXAMPLE
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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.)
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MAPLE
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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:
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MATHEMATICA
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{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 *)
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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