A048158 Triangular array T read by rows: T(n,k) = n mod k, for k=1,2,...,n, n=1,2,...

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 3, 2, 1, 0, 0, 0, 2, 0, 3, 2, 1, 0, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 2, 0, 5, 4, 3, 2, 1, 0, 0, 1, 1, 1, 3, 1, 6, 5, 4, 3, 2, 1, 0, 0, 0, 2, 2, 4, 2, 0, 6, 5, 4, 3, 2, 1, 0

Offset

1,13

Comments

Also, rectangular array read by antidiagonals: a(n, k) = n mod k, n >= 0, k >= 1. Cf. A051126, A051127, A051777. - David Wasserman, Oct 01 2008

Links

Alois P. Heinz, Rows n = 1..141, flattened

Michael Z. Spivey, The Humble Sum of Remainders Function, Mathematics Magazine, Vol. 78, No. 4 (Oct., 2005), pp. 300-305.

Eric Weisstein's World of Mathematics, Mod.

Formula

A051731(n,k) = A000007(T(n,k)). - Reinhard Zumkeller, Nov 01 2009

T(n,k) = n - k*A010766(n,k). - Mats Granvik, Gary W. Adamson, Feb 20 2010

Example

Triangle begins
  0;
  0  0;
  0  1  0;
  0  0  1  0;
  0  1  2  1  0;
  0  0  0  2  1  0;
  0  1  1  3  2  1  0;
  0  0  2  0  3  2  1  0;
  0  1  0  1  4  3  2  1  0;
  0  0  1  2  0  4  3  2  1  0;
  0  1  2  3  1  5  4  3  2  1  0;
  0  0  0  0  2  0  5  4  3  2  1  0;
  ...
From Omar E. Pol, Feb 21 2014: (Start)
Illustration of the 12th row of triangle:
-----------------------------------
.      k: 1 2 3 4 5 6 7 8 9 10..12
-----------------------------------
.         _ _ _ _ _ _ _ _ _ _ _ _
.        |_| | | | | | | | | | | |
.        |_|_| | | | | | | | | | |
.        |_| |_| | | | | | | | | |
.        |_|_| |_| | | | | | | | |
.        |_| | | |_| | | | | | | |
.        |_|_|_| | |_| | | | | | |
.        |_| | | | | |_| | | | | |
.        |_|_| |_| | |*|_| | | | |
.        |_| |_| | | |* *|_| | | |
.        |_|_| | |_| |* * *|_| | |
.        |_| | | |*| |* * * *|_| |
.        |_|_|_|_|*|_|* * * * *|_|
.
Row 12 is 0 0 0 0 2 0 5 4 3 2 1 0
.
(End)

Maple

T:= (n, k)-> modp(n, k):
seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Apr 04 2012

Mathematica

Flatten[Table[Mod[n, Range[n]], {n, 15}]]

Prog

(Haskell)
a048158 = mod
a048158_row n = a048158_tabl !! (n-1)
a048158_tabl = zipWith (map . mod) [1..] a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015, Jan 20 2014 (fixed), Aug 13 2013

Crossrefs

Row sums are given by A004125.

Cf. A002260.

Cf. A000007, A010766, A051126, A051127, A051731, A051777.

Sequence in context: A112207 A112208 A339089 * A275342 A246838 A219479

Adjacent sequences: A048155 A048156 A048157 * A048159 A048160 A048161

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

More terms from David Wasserman, Oct 01 2008

Status

approved