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A256285 a(n) = smallest divisor of the concatenation of n+1 and n, that did not occur earlier.

%I A256285 #21 Dec 23 2024 14:53:44
%S A256285 1,2,43,3,5,4,29,7,109,6,173,8,9,757,17,11,23,14,673,10,2221,18,2423,
%T A256285 631,15,47,257,12,13,313,359,28,3433,19,727,467,1279,22,577,20,4241,
%U A256285 26,1481,16,929,21,37,1237,27,25,59,24,41,2777,39,1439,5857,331,73
%N A256285 a(n) = smallest divisor of the concatenation of n+1 and n, that did not occur earlier.
%C A256285 Is this a permutation of the integers > 0 ?
%C A256285 From _Robert Israel_, May 20 2024: (Start)
%C A256285 Yes, this is a permutation of the positive integers.
%C A256285 For any positive integer k, there are arbitrarily large d such that 10^(d-1) > k and GCD(10^d + 1, k) == 1.  For such d, there is n such that  n == -10^d (10^d + 1)^(-1) (mod k) and 10^d > n >= 10^(d-1), and this implies that the concatenation of n+1 and n, which is 10^d * (n+1) + n, is divisible by k.  After all numbers < k have occurred, the next such n must have a(n) = k. (End)
%H A256285 Reinhard Zumkeller, <a href="/A256285/b256285.txt">Table of n, a(n) for n = 1..10000</a>
%H A256285 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-June/014948.html">Divisors of the concatenation of n+1 and n</a>, SeqFan list, Jun 03 2015.
%p A256285 R:= NULL: S:= {}:
%p A256285 for n from 1 to 100 do
%p A256285   v:=  10^(1+ilog10(n))*(n+1)+n;
%p A256285   s:= min(numtheory:-divisors(v) minus S);
%p A256285   R:= R,s;
%p A256285   S:= S union {s};
%p A256285 od:
%p A256285 R; # _Robert Israel_, May 20 2024
%o A256285 (Haskell)
%o A256285 import Data.List (insert); import Data.List.Ordered (minus)
%o A256285 a256285 n = a256285_list !! (n-1)
%o A256285 a256285_list = f (tail a127423_list) [] where
%o A256285    f (x:xs) ds = y : f xs (insert y ds) where
%o A256285                  y = head (a027750_row x `minus` ds)
%Y A256285 Cf. A027750, A127423, A253253.
%K A256285 nonn,base
%O A256285 1,2
%A A256285 _Eric Angelini_ and _Reinhard Zumkeller_, Jun 03 2015