[seqfan] Re: Fractions with no repeated digits.

[M. F. Hasler]

On Thu, Oct 10, 2019 at 5:07 AM David Corneth <davidacorneth at gmail.com>
wrote:

> Okay. You could maybe have a sequence T(m, k) giving the number of values t
> such that t*m and t*k share no digits in base 10.
> I don't know if that's interesting. Would you consider more bases?
>

With this definition, T(m,k) would be infinite for most small indices,
e.g., if  m & k  both have d digits, none in common, you have all  t =
sum(j=0,x, 10^(jd)), x >= 0.

In the spirit of Eric's idea, it would rather be :
T(m,k) = #{ t  s.t. concat(t*m, t*k) has no digit twice (or more)}
Then it's always finite since we must have concat(t*m, t*k) < 9876543210.
and  T(m,k) = T(k,m)  and  T(m,m) = 0  so one could/should impose k < m
and also T(m,k) = 0 whenever 10 | m and 10 | k, or m = k*10^z.

Yes, looking at other bases might be interesting. (Although probably dull
in very small bases, like 2.)

- Maximilian


> On Wed, Oct 9, 2019 at 7:17 PM Éric Angelini <bk263401 at skynet.be> wrote:
>
> > As MH wrote me in a private mail, the word "fraction"
> > is useless and misleading.
> > What I meant is {n,2n} doesn't show any repeated digit...
> > Sorry for that.
> > Best,
> > É.
> >
> >
> >
> >
> > > Le 8 octobre 2019 à 14:24, Éric Angelini <bk263401 at skynet.be> a écrit
> :
> > >
> > >
> > > Hello SeqFans,
> > > Here are the 304 integers such that the fraction n/2n repeats no digit:
> > >
> > > 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 19, 23, 27, 28, 29,
> > 31, 32, 34, 35, 36, 38, 39, 41, 43, 45, 46, 48, 52, 53, 54, 64, 65, 67,
> 69,
> > 73, 76, 78, 79, 82, 85, 86, 92, 93, 134, 135, 138, 139, 143, 145, 148,
> 152,
> > 154, 164, 176, 178, 179, 182, 185, 186, 192, 208, 209, 215, 218, 219,
> 235,
> > 238, 239, 267, 269, 273, 293, 307, 309, 314, 327, 329, 341, 345, 351,
> 352,
> > 354, 356, 358, 359, 364, 381, 382, 391, 392, 406, 413, 415, 416, 431,
> 435,
> > 436, 451, 453, 456, 458, 465, 476, 478, 481, 485, 486, 523, 532, 534,
> 536,
> > 538, 539, 543, 546, 548, 635, 639, 645, 652, 654, 679, 685, 692, 728,
> 729,
> > 764, 769, 782, 792, 793, 827, 835, 845, 852, 853, 865, 923, 927, 932,
> 935,
> > 936, 1345, 1354, 1435, 1453, 1465, 1478, 1485, 1546, 1548, 1645, 1728,
> > 1729, 1764, 1782, 1792, 1827, 1845, 1852, 1927, 2069, 2078, 2079, 2093,
> > 2178, 2179, 2185, 2309, 2358, 2359, 2708, 2709, 2718, 2719, 2807, 2817,
> > 2907, 2917, 3079, 3092, 3145, 3209, 3451, 3485, 3514, 3541, 3546, 3548,
> > 3564, 3582, 3592, 3609, 3645, 3845, 3906, 4076, 4135, 4176, 4351, 4356,
> > 4513, 4516, 4518, 4531, 4536, 4538, 4586, 4615, 4635, 4651, 4685, 4781,
> > 4815, 4835, 4851, 4853, 4856, 4865, 5238, 5239, 5364, 5382, 5392, 5436,
> > 5486, 6354, 6435, 6485, 6729, 6792, 6852, 6927, 7269, 7293, 7329, 7692,
> > 7923, 7932, 8235, 8352, 8523, 8532, 8546, 8645, 8652, 9235, 9267, 9273,
> > 9327, 9352, 13485, 13548, 13845, 14538, 14685, 14835, 14853, 14865,
> 15486,
> > 16485, 18546, 18645, 20679, 20769, 20793, 23079, 26709, 26907, 27069,
> > 27093, 27309, 29067, 29073, 29307, 30729, 30792, 30927, 31485, 32079,
> > 32709, 32907, 34851, 35148, 35481, 38145, 38451, 45138, 45186, 45381,
> > 46185, 46851, 48135, 48351, 48513, 48516, 48531, 48615, 48651.
> > >
> > > And here are the 153 integers such that the fraction n/3n repeats no
> > digit either:
> > >
> > > 1, 2, 3, 4, 6, 7, 8, 9, 12, 16, 18, 19, 21, 23, 26, 27, 29, 32, 34, 36,
> > 43, 46, 54, 58, 63, 64, 67, 68, 69, 73, 78, 81, 83, 87, 89, 91, 108, 109,
> > 126, 129, 136, 163, 168, 169, 176, 178, 182, 183, 189, 192, 194, 218,
> 219,
> > 236, 238, 246, 261, 263, 267, 269, 273, 291, 304, 306, 307, 308, 316,
> 318,
> > 326, 327, 354, 358, 364, 534, 543, 568, 582, 583, 594, 609, 634, 658,
> 673,
> > 678, 681, 683, 691, 768, 819, 839, 873, 891, 906, 916, 918, 1089, 1269,
> > 1609, 1678, 1683, 1694, 1736, 1746, 1763, 1768, 1782, 1794, 1809, 1823,
> > 1832, 1908, 1934, 2058, 2178, 2183, 2318, 2368, 2538, 2638, 2673, 2691,
> > 3054, 3058, 3087, 3168, 3176, 3182, 3218, 3267, 3582, 3594, 3658, 4609,
> > 5683, 5694, 5823, 5832, 5934, 6358, 6819, 6839, 6918, 8169, 8369, 9046,
> > 9136, 9168, 16794, 17694, 20583, 23058, 30582, 32058.
> > >
> > > Those examples suggest more of such sequences. For instance 20/19 would
> > produce the finite (and quite uninteresting, I have to admit) seq 20, 40,
> > 60, 80, 620, 920, 4160, 7320, 8360, 51360, 52380, 72540, 91320. We see
> > indeed that 20/19 = 91320/86754, both frations repeating no digits.
> > >
> > > Best,
> > > É.
> > >
>