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A258227 Concatenate the natural numbers, then partition into minimal strings so that adjacent terms have a common divisor greater than 1.

%I A258227 #17 Dec 04 2021 12:34:07
%S A258227 12,3,45,6,78,9,1011,12,1314,15,1617,18,192,0,2,12,2,2,32,4,2,52,6,2,
%T A258227 72,8,2,930,3,132,3,3,3,435,3,6,3,738,3,9,4041,42,4,34,4,4,54,6,4,74,
%U A258227 8,4,950,5,15,25,35,45,5,5,65,75,85,960,6,16,2,6,3,6
%N A258227 Concatenate the natural numbers, then partition into minimal strings so that adjacent terms have a common divisor greater than 1.
%C A258227 00 -> 0 is not allowed, else all digits will not appear in the concatenation of terms.  For example, a(198)..a(201) = 198, 19920, 0, 2 and not 198, 192, 0, 2. - _Michael S. Branicky_, Dec 03 2021
%H A258227 Michael S. Branicky, <a href="/A258227/b258227.txt">Table of n, a(n) for n = 1..10000</a> (24 terms corrected in terms 1..10000 from Reinhard Zumkeller)
%F A258227 GCD(a(n), a(n+1)) > 1.
%e A258227 .  a(n) | 12,3,45,6,78,9,1011,12,1314,15,1617,18,192,0,2,12,2,2,32,4,2,52
%e A258227 --------+----------------------------------------------------------------
%e A258227 .  gcd  |   3 3  3 6  3 3    3  6    3  3    3  6 192 2 2  2 2 2  4 2 2  .
%o A258227 (Haskell)
%o A258227 a258227 n = a258227_list !! (n-1)
%o A258227 a258227_list = f 12 1 (map toInteger $ tail a007376_list) where
%o A258227    f x y (d:ds) | gcd x y > 1 = y : f y d ds
%o A258227                 | otherwise   = f x (10 * y + d) ds
%o A258227 (Python)
%o A258227 from math import gcd
%o A258227 from itertools import count
%o A258227 def diggen():
%o A258227     for k in count(1): yield from list(map(int, str(k)))
%o A258227 def aupton(terms):
%o A258227     g = diggen()
%o A258227     alst, aset = [12], {12}
%o A258227     _, _, nxtd, nxtnxtd = next(g), next(g), next(g), next(g)
%o A258227     for n in range(2, terms+1):
%o A258227         an, nxtd, nxtnxtd = nxtd, nxtnxtd, next(g)
%o A258227         while gcd(an, alst[-1]) == 1 or nxtd == nxtnxtd == 0:
%o A258227             an, nxtd, nxtnxtd = int(str(an) + str(nxtd)), nxtnxtd, next(g)
%o A258227         alst.append(an); aset.add(an)
%o A258227     return alst
%o A258227 print(aupton(70)) # _Michael S. Branicky_, Dec 03 2021
%Y A258227 Cf. A002782, A007376.
%K A258227 nonn,base
%O A258227 1,1
%A A258227 _Reinhard Zumkeller_, May 23 2015