A390101 Irregular triangle read by rows: T(n, k) is the minimum positive x such that (n mod x) = k.

1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 5, 4, 1, 2, 5, 4, 1, 7, 3, 5, 1, 2, 7, 6, 5, 1, 3, 4, 7, 6, 1, 2, 3, 4, 7, 6, 1, 11, 5, 9, 8, 7, 1, 2, 11, 5, 9, 8, 7, 1, 13, 3, 11, 5, 9, 8, 1, 2, 13, 4, 11, 10, 9, 8, 1, 3, 7, 13, 6, 11, 10, 9, 1, 2, 3, 7, 13, 6, 11, 10, 9, 1, 17, 4, 5, 7, 13, 12, 11, 10

Offset

1,4

Links

Jason Yuen, Table of n, a(n) for n = 1..10100 (rows n = 1..200, flattened)

Formula

T(n, 0) = 1.

T(n, k) <= n-k for 1 <= k <= A110654(n)-1.

Example

Triangle begins:
  1;
  1;
  1, 2;
  1, 3;
  1, 2, 3;
  1, 5, 4;
  1, 2, 5, 4;
  1, 7, 3, 5;
  1, 2, 7, 6, 5;
  1, 3, 4, 7, 6;
  1, 2, 3, 4, 7, 6;
T(6, 1) = 5 because (6 mod 5) = 1 and there is no smaller solution to (6 mod x) = 1.

Mathematica

T[n_, k_]:=Module[{x=1}, While[Mod[n, x]!=k, x++]; x]; Table[T[n, k], {n, 18}, {k, 0, Ceiling[n/2]-1}]//Flatten (* Stefano Spezia, Oct 26 2025 *)

Prog

(Python)
def row(n):
    r = [-1]*(n+1>>1)
    for x in range(n, 0, -1): r[n%x] = x
    return r
print([x for n in range(1, 19) for x in row(n)])
(PARI) T(n, k) = for (i=1, n, if ((n % i) == k, return(i)));
row(n) = vector(ceil(n/2), i, T(n, i-1)); \\ Michel Marcus, Oct 25 2025

Crossrefs

Cf. A048158, A110654 (row length).

Columns 0..1: A000012, A020639.

Sequence in context: A279782 A358921 A379692 * A132589 A054843 A277427

Adjacent sequences: A390098 A390099 A390100 * A390102 A390103 A390104

Keyword

nonn,easy,tabf,look

Author

Jason Yuen, Oct 24 2025

Status

approved