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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)
OFFSET
1,1
COMMENTS
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
LINKS
EXAMPLE
230231232 is the concatenation of 230, 231 and 232 and is divisible by 231 and 232.
MAPLE
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]);
PROG
(Python)
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
CROSSREFS
Subsequence of A001703.
Cf. A308527.
Sequence in context: A360639 A227522 A193431 * A303241 A004945 A004965
KEYWORD
nonn,base,more
AUTHOR
J. M. Bergot and Robert Israel, Jun 03 2019
STATUS
approved

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Last modified February 28 07:05 EST 2024. Contains 370387 sequences. (Running on oeis4.)