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%I #9 Oct 18 2017 16:30:20
%S 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,
%T 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,
%U 1,2,5,15,25,75,125,175,225,1,2,5,15,25,75,125,175,225
%N 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)).
%C 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.
%C 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?
%e The table starts as follows:
%e n | N=A292683(n) | N/A217657(N) | A292685(n,k=1..A292684(n))
%e 1 | 11 | 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9
%e 2 | 12 | 6 | 1, 2, 3, 4
%e 3 | 15 | 3 | 1
%e 4 | 21 | 21 | 1, 2, 3, 4, 5, 15, 25, 35, 45
%e 5 | 24 | 6 | 1, 2, 5, 15
%e | (...) | (...) | (...)
%e 68 | 416 | 26 | 1, 2, 5, 15, 25, 75, 125, 175, 225
%e (...)
%e 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).
%e 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.
%e 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.
%o (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).
%Y Cf. A292683, A292684 (gives the row lengths), A217657, A000030.
%K nonn,base,tabf
%O 1,2
%A _M. F. Hasler_, Oct 18 2017
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