A292685 Irregular table where row n lists the positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 5, 15, 1, 5, 15, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225

Offset

1,2

Comments

The numbers N listed in A292683 are such that N is divisible by A217657(N) = N with its initial digit removed. Most of these numbers have several multiples k*N which again have this property, but furthermore, such that the ratio k*N / A217657(k*N) is always the same. The corresponding k-values are listed here.

It is not rare that there are 9 such k-values, although the set of these is usually different from { 1, ..., 9 }. Is there any N for which there more than 10 such k-values?

Links

Table of n, a(n) for n=1..77.

Example

The table starts as follows:
    n | N=A292683(n) | N/A217657(N) | A292685(n,k=1..A292684(n))
    1 |      11      |      11      | 1, 2, 3, 4, 5, 6, 7, 8, 9
    2 |      12      |       6      | 1, 2, 3, 4
    3 |      15      |       3      | 1
    4 |      21      |      21      | 1, 2, 3, 4, 5, 15, 25, 35, 45
    5 |      24      |       6      | 1, 2, 5, 15
      |    (...)     |    (...)     | (...)
   68 |     416      |      26      | 1, 2, 5, 15, 25, 75, 125, 175, 225
           (...)
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6, e.g., 12*4 = 48, 48 / 8 = 6 (= 12 / 2).
There are other k such that 12*k is divisible by A217657(12*k), e.g., k = 6, 7, 8, 17, ... (=> 12*k = 72, 84, 96, 204: all divisible by their last digit), but which yield ratios (here 36, 21, 16, 51) different from 6.
For n = 4, we have, e.g., 21*15 = 315, 315 / 15 = 21 (= 21 / 1), or 21*45 = 945, 945 / 45 = 21. Here too, e.g., 21*24 = 504 is divisible by 04, but 504 / 4 = 126, not 21.

Prog

(PARI) A292685_row(n, N=A292683(n), r=N/A217657(N), a=[1])={for(k=2, oo, if(k%10, A217657(k*N)*r==k*N&&a=concat(a, k), k<10*a[#a]||break)); a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. It is not checked whether N is in A292683 (else the resulting vector should be empty).

Crossrefs

Cf. A292683, A292684 (gives the row lengths), A217657, A000030.

Sequence in context: A180410 A071650 A037265 * A285093 A053837 A128244

Adjacent sequences: A292682 A292683 A292684 * A292686 A292687 A292688

Keyword

nonn,base,tabf

Author

M. F. Hasler, Oct 18 2017

Status

approved