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A306643 Numbers that, for some x, are the concatenation of x, x+1 and x+2 and are divisible by at least two of x, x+1 and x+2. 1
123, 234, 345, 456, 8910, 101112, 230231232 (list; graph; refs; listen; history; text; internal format)



012 is not included because leading 0's are not allowed.

From Charlie Neder, Jun 05 2019: (Start)

If x = 10^k - 2, then x|x+1|x+2 (with | denoting concatenation) will be congruent to 22 modulo x, -9 modulo x+1, and 0 modulo x+2.

If x = 10^k - 1, then x|x+1|x+2 will be congruent to 12 modulo x and 1 modulo x+1.

Therefore, the only term such that x and x+2 have different lengths is 8910.

By reducing modulo x + {0,1,2} it can be shown that if at least two of x|10^k+2, x+1|10^2k-1, and x+2|2*10^2k-10^k are true - presuming x and x+2 are the same length - then x|x+1|x+2 is in this sequence. No further terms corresponding to x < 10^18. (End)

No further terms corresponding to x < 10^50. - Chai Wah Wu, Jun 19 2019


Table of n, a(n) for n=1..7.


230231232 is the concatenation of 230, 231 and 232 and is divisible by 231 and 232.


cat3:= proc(x)

  local t;

  t:= 10^length(x+2);

  x*(1 + t*(1+10^length(x+1)))+t+2

end proc:

f:= proc(x) local q, a, b;

  q:= cat3(x);

  a:= (q/x)::integer;

  b:= (q/(x+1))::integer;

  if a and b then return q elif not(a) and not(b) then return NULL fi;

  if (q/(x+2))::integer then q else NULL fi

end proc:

map(f, [$1..1000]);



for k in range(1, 8):

..for x in range(10**(k-1), 10**k-2): # will not find 8910

....if sum([not (10**k+2)%x, not (10**(2*k)-1)%(x+1), \

....not (2*10**(2*k)+10**k)%(x+2)]) >= 2:

......print(str(x)+str(x+1)+str(x+2)) # Charlie Neder, Jun 05 2019


Subsequence of A001703.

Cf. A308527.

Sequence in context: A153254 A227522 A193431 * A303241 A004945 A004965

Adjacent sequences:  A306640 A306641 A306642 * A306644 A306645 A306646




J. M. Bergot and Robert Israel, Jun 03 2019



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Last modified September 23 02:41 EDT 2021. Contains 347609 sequences. (Running on oeis4.)