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A253253 a(n) = smallest divisor of the concatenation of n and n+1 that did not occur earlier. 2
1, 23, 2, 3, 4, 67, 6, 89, 5, 337, 8, 1213, 9, 283, 379, 7, 859, 17, 10, 43, 1061, 13, 14, 25, 421, 37, 11, 41, 293, 433, 12, 53, 1667, 15, 16, 3637, 21, 349, 20, 449, 19, 4243, 24, 35, 2273, 1549, 1187, 373, 18, 5051, 28, 51, 2677, 1091, 463, 5657, 2879, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Is this a permutation of the integers > 0 ?

Comment from N. J. A. Sloane, May 19 2017 (Start):

It should not be difficult to prove that every positive integer appears.

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

m divides the concatenation of N and N+1.

Let N have k digits and suppose that

10^(k-1) <= N <= 10^k - 2.

The concatenation of N and N+1 is N*(10^k+1)+1, so we want to find numbers k and N such that

N*(10^k+1) == -1 mod m.

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.

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.

The other cases are left to the reader. (End)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

√Čric Angelini, Divisors of the concatenation of n and n+1, SeqFan list, Jun 03 2015.

PROG

(Haskell)

import Data.List (insert); import Data.List.Ordered (minus)

a253253 n = a253253_list !! (n-1)

a253253_list = f a001704_list [] where

   f (x:xs) ds = y : f xs (insert y ds) where

                 y = head (a027750_row' x `minus` ds)

CROSSREFS

Cf. A027750, A001704, A256285.

Sequence in context: A040521 A040524 A306374 * A225003 A040525 A040526

Adjacent sequences:  A253250 A253251 A253252 * A253254 A253255 A253256

KEYWORD

nonn,base,look

AUTHOR

Eric Angelini and Reinhard Zumkeller, Jun 05 2015

STATUS

approved

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)