cp's OEIS Frontend

A249647 Numbers n that divide the concatenation of n+1 and n-1.

%I A249647 #31 Jun 11 2016 01:02:29
%S A249647 1,3,9,11,33,99,111,333,999,1111,3333,9999,11111,33333,99999,111111,
%T A249647 142857,333333,999999,1111111,3333333,9999999,11111111,33333333,
%U A249647 99999999,111111111
%N A249647 Numbers n that divide the concatenation of n+1 and n-1.
%C A249647 A069871 is a subset of this sequence.
%C A249647 All the numbers of the form (10^k - 1)/9, (10^k - 1)/3 and (10^k-1) are members.
%C A249647 Any concatenation of 142857 with itself is part of the sequence, too.
%C A249647 All integers of the form (10^d-1)/k where 1 <= k <= 9. - _Robert Israel_, Jan 06 2016
%C A249647 It appears that equivalent definitions are: (A) Numbers which divide the repunit of the same length, n | R(length(n)). (B) Numbers equal to one or more concatenations of one among {1, 3, 9, 142857}. Is there a proof for this? - _M. F. Hasler_, Jun 10 2016
%H A249647 Robert Israel, <a href="/A249647/b249647.txt">Table of n, a(n) for n = 1..3163</a>
%F A249647 a(n) = 1000001*a(n-19) + 1000000*a(n-38). - _Robert Israel_, Jan 06 2016
%e A249647 3 belongs to this sequence as 3 divides 42, 11 belongs to this sequence as 11 divides 1210.
%e A249647 9 belongs to this sequence as 9 divides the concatenation of 10 and 8, i.e., 108.
%p A249647 with(numtheory): P:=proc(q) local n;
%p A249647 for n from 1 to q do if type(((n+1)*10^(ilog10(n-1)+1)+n-1)/n,integer) then print(n);
%p A249647 fi; od; end: P(10^15);
%p A249647 # alternative:
%p A249647 seq(op(select(type, [seq((10^d-1)/k, k=[9,7,3,1])],integer)),d=1..20); # _Robert Israel_, Jan 06 2016
%t A249647 Select[Range[2, 10^7], Divisible[FromDigits[IntegerDigits[# + 1]~Join~IntegerDigits[# - 1]], #] &] (* _Michael De Vlieger_, Jan 06 2016 *)
%o A249647 (PARI) for(n=1,10^8, s=concat(Str(n+1),Str(n-1));if(!(eval(s)%n),print1(n,", "))) \\ _Derek Orr_, Nov 03 2014
%Y A249647 Cf. A069871.
%K A249647 nonn,base
%O A249647 1,2
%A A249647 _Paolo P. Lava_, Nov 03 2014
%E A249647 a(23)-a(25) from _Derek Orr_, Nov 03 2014