A375117 Irregular triangle of positive integers, read by rows, the elements of the n-th row being the nonzero remainders, in increasing order, when the Euclidean algorithm is applied to 2^n-1 and n.
1, 1, 1, 3, 1, 3, 1, 1, 7, 1, 2, 7, 1, 3, 1, 3, 1, 1, 2, 3, 1, 7, 1, 15, 1, 9, 1, 5, 15, 7, 1, 3, 1, 3, 6, 9, 15, 1, 6, 1, 2, 3, 1, 2, 25, 1, 2, 13, 15, 1, 3, 1, 1, 31, 1, 2, 5, 7, 1, 3, 1, 17, 9, 27, 1, 1, 2, 3, 1, 3, 4, 7, 5, 10, 15, 1, 21, 1, 1, 14, 15, 1, 3, 13, 16
Offset: 2
Examples
The triangle begins:
1;
1;
1, 3;
1;
3;
1;
1, 7;
1, 2, 7;
1, 3;
1;
3;
1;
3;
...
Row(2) is {1}, because 2^2-1 = 4-1 = 3, and 3 divided by 2 leaves a remainder of 1.
Row(4) is {1, 3}, because 2^4-1 = 16-1 = 15, and 15 divided by 4 leaves a remainder of 3, and 4 divided by 3 leaves a remainder of 1.
Programs
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PARI
row(n) = my(x=2^n-1, y=n, ok=1, list=List()); while (ok, my(z=divrem(x, y)); x = y; y = z[2]; if (y==0, ok=0, listput(list, y));); listsort(list); Vec(list); \\ Michel Marcus, Jul 31 2024
Extensions
More terms from Michel Marcus, Jul 31 2024