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A088797 Numbers n > 2 such that n divides the concatenation of n-2 and n-1. 15
3, 7, 67, 667, 6667, 66667, 666667, 2857143, 6666667, 66666667, 666666667, 1052631579, 6666666667, 66666666667, 666666666667, 2857142857143, 6666666666667, 11764705882353, 66666666666667, 666666666666667 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For a 10-digit number, the difference between cat((n+2),(n+1)) and cat((n-2),(n-1)) is 40000000002 (as long as n-2 to n+2 are all numbers with the same number of digits). This difference has only 3 divisors which are ten digits long (1052631579, 2105263158 and 6666666667) of which two belong to the sequence. As 40000000002 has no other 10-digit factors, it is necessary to consider 11-digit numbers to obtain more terms.

From Robert G. Wilson v, Oct 21 2003, Oct 28 2003, Sep 23 2015 & Oct 24 2015: (Start)

All numbers of the forms

2(10^n-1)/3 + 1,

floor(2(10^(6n + 1) - 1)/7 + 1),

floor(2(10^(16n - 2) - 1)/17 + 1), and

floor(2(10^(18n - 8) - 1)/19 + 1), for n > 0 are members.

The only term not one of the above forms so far is 3. But it is included when n=0 for the second form.

(End)

If numbers less than 3 are acceptable, then an argument could be made that 1 is a terms since cat(n-2,n-1) is -10 which is == 0 (mod 1). - Robert G. Wilson v, Sep 29 2015

From Robert Israel, Oct 18 2015: (Start)

Numbers n of the form (2*10^m + 1)/k where k = 3, or k = 7 and m == 1 mod 6, or k = 17 and m == 14 mod 16, or k = 19 and m == 10 mod 18.

This is because n | (n-2)*10^m + (n-1) iff n | 2*10^m + 1.

But since we need 10^m >= n > 10^(m-1), 2*10^m+1 = k*n where 3 <= k <= 20.

The only numbers in that range that ever divide 2*10^m+1 are 3,7,17,19. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 1..1285 (n = 1..104 from Robert G. Wilson v)

EXAMPLE

a(2)=7 because (7-2) concatenated with (7-1) yields 56 and 7 is a divisor of 56.

a(4)=667 because 667 is a divisor of 665666.

.

MAPLE

M:= 20: # to get all terms with <= M digits

select(type, [seq(seq((2*10^m+1)/k, k=[19, 17, 7, 3]), m=1..M)], integer); # Robert Israel, Oct 18 2015

MATHEMATICA

Select[ Range[8250000000], Mod[ FromDigits[ Join[ IntegerDigits[ # - 2], IntegerDigits[ # - 1]]], # ] == 0 &]

fQ[n_] := Mod[ FromDigits[ Join[ IntegerDigits[n - 2], IntegerDigits[n - 1]]], n] == 0; k = 1; lst = {}; Select[ Flatten@ Table[ Select[ Divisors[4*10^n + 2], 10^(n - 1) < # < 10^n &], {n, 15}], fQ] (* Robert G. Wilson v, Sep 05 2015 *)

PROG

(PARI) for(n=3, 1e6, if((k=eval(Str(n-2, n-1))) && k % n == 0, print1(n", "))) \\ Altug Alkan, Sep 25 2015

CROSSREFS

Cf. A069860, A069862, A069871, A088798.

Sequence in context: A219845 A041817 A120364 * A165589 A184316 A127177

Adjacent sequences:  A088794 A088795 A088796 * A088798 A088799 A088800

KEYWORD

base,nonn

AUTHOR

Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 18 2003

EXTENSIONS

Extended by Robert G. Wilson v, Oct 21 2003

Further terms from Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 29 2003

STATUS

approved

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Last modified July 18 11:39 EDT 2019. Contains 325139 sequences. (Running on oeis4.)