A060284 -id:A060284 - cp's OEIS frontend

A383188 Irregular table, read by rows, where row z = 2, 3, 4, ... lists pairs (y, x) such that x + y/z = concat(y, x)/z with 0 < y < z, gcd(y, z) = 1, and primitive x, cf. comments.

1, 9, 2, 9, 1, 3, 3, 9, 4, 9, 5, 9, 2, 3, 4, 6, 6, 9, 1, 142857, 3, 428571, 5, 714285, 7, 9, 8, 9, 3, 3, 7, 7, 9, 9, 10, 9, 5, 45, 7, 63, 11, 9, 4, 3, 8, 6, 12, 9, 3, 230769, 5, 384615, 7, 538461, 11, 846153, 13, 9, 2, 142857, 4, 285714, 8, 571428, 14, 9, 5, 3, 15, 9, 16, 9, 5, 2941176470588235, 7, 4117647058823529, 11, 6470588235294117, 13, 7647058823529411, 17, 9, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, 14, 7, 16, 8, 18, 9

Offset: 2

M. F. Hasler, May 03 2025

nonn

The numbers x + y/z = concat(y,x) / z have been called "Harry Potter numbers" by Romy Aran, cf. link to SeqFan post.
concat(y, x) := y*10^Lx + x, with Lx = floor(log_10(x))+1 = A055642(x) = number of decimal digits of x.
For any solution (x, y, z), any triple (x*{k}, y, z) is again a solution, where x*{k} = concat(x*{k-1}, x), x*{1} = x. Therefore we consider only "primitive" x, which are not of this form with some k > 1.
For any z there is the solution (x, y, z) = (9, z-1, z), which is usually listed last in each row, so the '9's preceded by y = z-1 are convenient "end-of-line" markers for reading the flattened sequence as a table.

			For any z > 1, we have the solution (x, y) = (9, z-1), and well as non-primitive x = 99, 999, ... (not listed in the table), for example:
z = 2, y = 1: 9 + 1/2 = 19/2, and 99 + 1/2 = 199/2, 999 + 1/2 = 1999/2, ...
z = 3, y = 2: 9 + 2/3 = 29/3, and 99 + 2/3 = 299/3, 999 + 2/3 = 2999/3, ...
z = 4, y = 3: 9 + 3/4 = 39/4, and 99 + 3/4 = 399/4, 999 + 3/4 = 3999/4, ...
But in row z = 4 we first list the solution y = 1, x = 3, viz: 3 + 1/4 = 13/4 (and 3...3 + 1/4 = 13...3 / 4).
In row z = 7 we first list the solution y = 2, x = 3, viz: 3...3 + 2/7 = 23...3 / 7, then y = 4, x = 6, viz: 6...6 + 4/7 = 46...6 / 7.
In row 8, we have the solutions 142857...142857 + 1/8 = 1142857...142857 / 8, 428571...428571 + 3/8 = 3428571...428571 / 8, 714285...714285 + 5/8 = 5714285...714285 / 8, 9...9 + 7/8 = 79...9 / 8.
The table starts:
   z  |  pairs (y, x) (= sequence data)
------+--------------------------------
   2  |  1, 9
   3  |  2, 9
   4  |  1, 3; 3, 9   (representing 3...3 + 1/4, and 9...9 + 3/4)
   5  |  4, 9
   6  |  5, 9
   7  |  2, 3; 4, 6; 6, 9
   8  |  1, 142857; 3, 428571; 5, 714285; 7, 9
   9  |  8, 9
  10  |  3, 3; 7, 7; 9, 9   (representing 3...3 + 3/10 = 33...3 / 10, etc.)
  11  |  10, 9              (representing 9...9 + 10/11 = 109...9 / 11)
  12  |  5, 45; 7, 63; 11, 9   (e.g., 63...63 + 7/12 = 763...63 / 12, etc.)
  13  |  4, 3; 8, 6; 12, 9
  14  |  3, 230769; 5, 384615; 7, 538461; 11, 846153; 13, 9
  15  |  2, 142857; 4, 285714; 8, 571428; 14, 9
  16  |  5, 3; 15, 9
  17  |  16, 9
  18  |  5, 2941176470588235; 7, 4117647058823529; 11, 6470588235294117;
      |  13, 7647058823529411; 17, 9
  19  |  2, 1; 4, 2; 6, 3; 8, 4; 10, 5; 12, 6; 14, 7; 16, 8; 18, 9

Romy Aran, Harry Potter numbers, post to the SeqFan google group, May 1, 2025.

Cf. A036275 and A060284 (periodic part of the decimal expansion of 1/n).

/* brute-force search, for illustration */
row(z, L=99, S=[]) = { for ( y = max(1, (z-1)\10+1), z-1, gcd(y, z) > 1 && next;
   my ( zz = Mod(10,z-1) ); for ( Lx = 1, L, y*(zz^Lx-1) && next;
   my ( x = y*(10^Lx-1)/(z-1) ); logint(x, 10)+1 == Lx || next;
   foreach ( S, xx, x%xx[2] && next;
      x\xx[2] == 10^Lx\(10^(logint(xx[2], 10)+1)-1) && next(2));
   S = concat(S,[[y, x]]) )/*for Lx*/ )/*for y*/; concat(S) }

cp's OEIS Frontend

A383188 Irregular table, read by rows, where row z = 2, 3, 4, ... lists pairs (y, x) such that x + y/z = concat(y, x)/z with 0 < y < z, gcd(y, z) = 1, and primitive x, cf. comments.

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