A069871 Numbers n that divide the concatenation of n-1 and n+1.

3, 9, 11, 33, 111, 333, 1111, 3333, 11111, 33333, 111111, 142857, 333333, 1111111, 3333333, 11111111, 33333333, 111111111, 333333333, 1111111111, 3333333333, 11111111111, 33333333333, 111111111111, 142857142857, 333333333333, 1111111111111

Offset

1,1

Comments

All the numbers of the form (10^k - 1)/3 and (10^k - 1)/9 are members.

Any concatenation of 142857 with itself is part of the sequence too. - Paolo P. Lava, Nov 03 2014

These (i.e. (10^k - 1)/3 for k>=1, (10^k - 1)/9 for k>=2, and (10^(6*k) - 1)/7 for k>=1) are all the members of the sequence, apart from 9. This is because if 10^(i-1) <= x+1 < 10^i, x | 10^i*(x-1) + x + 1 iff x | 10^i - 1, and then 1 < d = (10^i - 1)/x <= (10^i - 1)/(10^(i-1)-1) < 10. Since 2,4,5,6,8 can't divide 10^i-1, d must be 3, 7 or 9. - Robert Israel, Nov 04 2014

The terms of the sequence satisfy the condition that both n-1 and n+1 must be greater than 0. If n-1=0 were admitted then 1 would also be part of the sequence. - Michel Marcus, Nov 05 2014

Links

Table of n, a(n) for n=1..27.

Formula

From Robert Israel, Nov 04 2014: (Start)

a(1+2*j + 13*k) = (10^(1+j+6*k)-1)/9, j=0..5, k >= 0 (except for j=k=0).

a(2*j + 13*k) = (10^(j+6*k)-1)/3, j=0..5, k >= 0 (except for j=k=0 and j=1,k=0).

a(13*k - 1) = (10^(6*k)-1)/7, k >= 1.

(End)

Example

3 belongs to this sequence since 3 divides 24; 11 belongs to this sequence since 11 divides 1012.
9 belongs to this sequence since 9 divides the concatenation of 8 and 10, i.e. 810.
142857 belongs to this sequence since 142857 divides the concatenation of 142856 and 142858, i.e. 142856142858/142857 = 999994.

Maple

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if type(((n-1)*10^(ilog10(n+1)+1)+n+1)/n, integer)
then print(n); fi; od; end: P(10^15); # Paolo P. Lava, Nov 03 2014
# Alternative:
N:= 10: # to get all terms with at most N digits
3, 9, seq(seq((10^k-1)/d, d = `if`(k mod 6 = 0, [9, 7, 3], [9, 3])), k = 2 .. N); # Robert Israel, Nov 04 2014

Mathematica

Select[ Range[10^8], Mod[ FromDigits[ Join[ IntegerDigits[ # - 1], IntegerDigits[ # + 1]]], # ] == 0 & ]

Prog

(PARI) isok(n) = eval(concat(Str(n-1), Str(n+1))) % n == 0; \\ Michel Marcus, Nov 04 2014

Crossrefs

Cf. A069860, A069862, A088797, A088798.

Cf. A077192.

Sequence in context: A106305 A135365 A088877 * A061957 A136984 A168400

Adjacent sequences: A069868 A069869 A069870 * A069872 A069873 A069874

Keyword

nonn,base

Author

Amarnath Murthy, Apr 24 2002

Extensions

More terms from Sascha Kurz, Feb 10 2003

Added missing a(12) by Paolo P. Lava and missing a(25) by Alois P. Heinz, Nov 03 2014

Status

approved