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

12, 3, 45, 6, 78, 9, 1011, 12, 1314, 15, 1617, 18, 192, 0, 2, 12, 2, 2, 32, 4, 2, 52, 6, 2, 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, 8, 4, 950, 5, 15, 25, 35, 45, 5, 5, 65, 75, 85, 960, 6, 16, 2, 6, 3, 6

Offset

1,1

Comments

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (24 terms corrected in terms 1..10000 from Reinhard Zumkeller)

Formula

GCD(a(n), a(n+1)) > 1.

Example

.  a(n) | 12,3,45,6,78,9,1011,12,1314,15,1617,18,192,0,2,12,2,2,32,4,2,52
--------+----------------------------------------------------------------
.  gcd  |   3 3  3 6  3 3    3  6    3  3    3  6 192 2 2  2 2 2  4 2 2  .

Prog

(Haskell)
a258227 n = a258227_list !! (n-1)
a258227_list = f 12 1 (map toInteger $ tail a007376_list) where
   f x y (d:ds) | gcd x y > 1 = y : f y d ds
                | otherwise   = f x (10 * y + d) ds
(Python)
from math import gcd
from itertools import count
def diggen():
    for k in count(1): yield from list(map(int, str(k)))
def aupton(terms):
    g = diggen()
    alst, aset = [12], {12}
    _, _, nxtd, nxtnxtd = next(g), next(g), next(g), next(g)
    for n in range(2, terms+1):
        an, nxtd, nxtnxtd = nxtd, nxtnxtd, next(g)
        while gcd(an, alst[-1]) == 1 or nxtd == nxtnxtd == 0:
            an, nxtd, nxtnxtd = int(str(an) + str(nxtd)), nxtnxtd, next(g)
        alst.append(an); aset.add(an)
    return alst
print(aupton(70)) # Michael S. Branicky, Dec 03 2021

Crossrefs

Cf. A002782, A007376.

Sequence in context: A317312 A112033 A248171 * A130895 A367431 A038329

Adjacent sequences: A258224 A258225 A258226 * A258228 A258229 A258230

Keyword

nonn,base

Author

Reinhard Zumkeller, May 23 2015

Status

approved