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A253253 a(n) = smallest divisor of the concatenation of n and n+1 that did not occur earlier.

%I A253253 #20 Dec 23 2024 14:53:44
%S A253253 1,23,2,3,4,67,6,89,5,337,8,1213,9,283,379,7,859,17,10,43,1061,13,14,
%T A253253 25,421,37,11,41,293,433,12,53,1667,15,16,3637,21,349,20,449,19,4243,
%U A253253 24,35,2273,1549,1187,373,18,5051,28,51,2677,1091,463,5657,2879,27
%N A253253 a(n) = smallest divisor of the concatenation of n and n+1 that did not occur earlier.
%C A253253 Is this a permutation of the integers > 0?
%C A253253 Comment from _N. J. A. Sloane_, May 19 2017: (Start)
%C A253253 It should not be difficult to prove that every positive integer appears.
%C A253253 If not, let m be the smallest missing number. There is an n_0 such that for all n >= n_0, a(n) > m. The theorem will follow if we can find an N > n_0 such that
%C A253253 m divides the concatenation of N and N+1.
%C A253253 Let N have k digits and suppose that
%C A253253 10^(k-1) <= N <= 10^k - 2.
%C A253253 The concatenation of N and N+1 is N*(10^k+1)+1, so we want to find numbers k and N such that
%C A253253 N*(10^k+1) == -1 mod m.
%C A253253 Case (i). If gcd(m,10)=1, then by Euler's theorem, 10^phi(m) == 1 mod m, so we can take k to be a sufficiently large multiple of phi(m), and then take N to be a number of the form r*m-1 in the range 10^(k-1) <= N <= 10^k - 2.
%C A253253 Case (ii). If m = 2^r or 5^r, then for large k, 10^k+1 == 1 mod m, and we take N to be of the form m*s-1 in the range 10^(k-1) <= N <= 10^k - 2.
%C A253253 The other cases are left to the reader. (End)
%H A253253 Reinhard Zumkeller, <a href="/A253253/b253253.txt">Table of n, a(n) for n = 1..10000</a>
%H A253253 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-June/014947.html">Divisors of the concatenation of n and n+1</a>, SeqFan list, Jun 03 2015.
%o A253253 (Haskell)
%o A253253 import Data.List (insert); import Data.List.Ordered (minus)
%o A253253 a253253 n = a253253_list !! (n-1)
%o A253253 a253253_list = f a001704_list [] where
%o A253253    f (x:xs) ds = y : f xs (insert y ds) where
%o A253253                  y = head (a027750_row' x `minus` ds)
%Y A253253 Cf. A027750, A001704, A256285.
%K A253253 nonn,base,look
%O A253253 1,2
%A A253253 _Eric Angelini_ and _Reinhard Zumkeller_, Jun 05 2015