**Share No Digit**

Hello SeqFans, [http://list.seqfan.eu]

We consider two
consecutive integers ("a" and "b") from a monotonically
increasing sequence **S**, and their sum
"c"; we want that:

"a" shares no digit with "b"

"b" shares no digit with "c"

"c" shares no digit with "a"

Starting with 1,
do we have:

**S** = 1,2,3,4,5,6,7,8,9,11,22,33,42,51,249,302,449,... ?

(to prolong **S**,
take the smallest integer not leading to a contradiction)

Does **S** stop at some point?

Best,

É.

[August, 31^{st},
2009]

_________

**Zak Seidov**:

Eric,

Your
sequence terminates at a(38)=5138383:

**S** = 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 42,
51, 249, 306, 419, 2587, 3413, 5255, 6064, 7255, 8844, 11156, 22222, 31778,
60444, 72755, 88344, 111656, 222088, 333361, 422224, 508887, 622224, 708887,
922224, 5138383.

_________

**Jack Brennen**:

Brings
up some interesting side questions, such as whether there is a bound on the **ratio** a(n+1)/a(n).

[Assume
that a(n) can be arbitrary; it doesn’t have to be
reached from a smaller number.]

For
instance, if a(n) is 13427, a(n+1) is 886668, for a
ratio of just over 66. How high could this ratio get?

_________

**Franklin T. Adams-Watters**

The next term can
have at most one digit more, so 100 is an upper bound.

_________

**Zak Seidov**:

Running
records in r=a(n+1)/a(n)

{a(n),a(n+1),r}

{1,2,2.}

{12,33,2.75}

{51,249,4.88235}

{415,2622,6.31807}

{512,3888,7.59375}

{513,6489,12.6491}

{1479,28826,19.4902}

{2934,77116,26.2836}

{11742,588558,50.1242}

{13427,886668,66.0362}

... Next record?

_________

**Franklin T. Adams-Watters**

I’ve been looking
at this in other bases. In bases 7 and 8, the sequence is infinite.

If my program is
correct, the base 7
sequence starts:

**S** = 1,2,3,4,5,6,8,16,24,30,40,60,114,171,212,285,403,800,1200,1486,2000,2804,5602,8403,10404,14005,19611,39216,58824,72830,98040

... which in base 7 is:

1,2,3,4,5,6,

11,22,33,42,55,

114,222,333,422,555,

1114,2222,3333,4222,5555,

11114,22222,33333,42222,55555,

111114,222222,333333,422222,555555

Base 8 starts:

**S** = 1,2,3,4,5,6,7,9,18,27,34,45,50,59,73,146,206,260,341,388,471,1609,2340,2706,3127,6235,21845,24836,30167,102985,149796,173202,200119,399067,1398101,1589508,1930711,6591049,9586980,11084946,12807607,25540315,89478485,101728516,123565527,

... which in base 8 is:

1,2,3,4,5,6,7,

11,22,33,42,55,62,73,

111,222,316,404,525,604,727,

3111,4444,5222,6067,

14133,52525,60404,72727,

311111,444444,522222,606667,

1413333,5252525,6040404,7272727,

31111111,44444444,52222222,60666667,

141333333,525252525,604040404,727272727

In both cases the
pattern is obvious.

These are the only
bases up to 10 where this happens.

In bases 2 and 3,
the sequence is only the number 1.

Base 4 gives us:

**S** = 1,2,3,5,10 (1,2,3,11,22).

Base 5:

**S** = 1,2,3,4,6,12,18,34 (1,2,3,4,11,22,33,114)

Base 6:

**S** = 1,2,3,4,5,7,14,21,26 (1,2,3,4,5,11,22,33,42)

Base 9:

**S** = 1,2,3,4,5,6,7,8,10,20,30,38,50,57,65,75,91,182,259,409,546,587,656,984,1932

(1,2,3,4,5,6,7,8,

11,22,33,42,55,63,72,83,

111,222,317,504,666,722,808,

1313,2576)

And base 10 (as others have
noted):

**S** = 1,2,3,4,5,6,7,8,9,11,22,33,42,51,249,306,419,2587,3413,5255,6064,7255,8844,11156,22222,31778,60444,72755,88344,111656,222088,333361,422224,508887,622224,708887,922224,5138383.

_________