A373917 Triangle read by rows: T(n,k) = k*10 mod n, with n >= 1, k >= 0.

0, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 4, 2, 0, 3, 6, 2, 5, 1, 4, 0, 2, 4, 6, 0, 2, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 8, 6, 4, 2, 0, 10, 8, 6, 4, 2, 0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3

Offset: 1

Author

Paolo Xausa, Jun 26 2024

Keywords

• nonn
• tabl

Comments

Each row n encodes a "division graph" used to determine m mod n (where m is an arbitrary nonnegative integer), using the method described in the Numberphile link (see also example).

Examples

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9
  ---------------------------------
   1 | 0;
   2 | 0, 0;
   3 | 0, 1, 2;
   4 | 0, 2, 0, 2;
   5 | 0, 0, 0, 0, 0;
   6 | 0, 4, 2, 0, 4, 2;
   7 | 0, 3, 6, 2, 5, 1, 4;
   8 | 0, 2, 4, 6, 0, 2, 4, 6;
   9 | 0, 1, 2, 3, 4, 5, 6, 7, 8;
  10 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  ...
Suppose m = 3714289 and you want to determine m mod 7 (the example shown in the video).
Start with the first digit of m (3) and calculate T(7,3 mod 7) = T(7,3) = 2.
Add it to the next digit of m (7) and calculate T(7,(2+7) mod 7) = T(7,2) = 6.
Add it to the next digit of m (1) and calculate T(7,(6+1) mod 7) = T(7,0) = 0.
Add it to the next digit of m (4) and calculate T(7,(0+4) mod 7) = T(7,4) = 5.
Add it to the next digit of m (2) and calculate T(7,(5+2) mod 7) = T(7,0) = 0.
Add it to the next digit of m (8) and calculate T(7,(0+8) mod 7) = T(7,1) = 3.
Add it to the final digit of m (9) and calculate (3+9) mod 7 = 5, which corresponds to 3714289 mod 7.
		

Links

• John Tyler Rascoe, Rows n = 1..150, flattened
• James Grime and Brady Haran, Solving Seven, Numberphile YouTube video, 2024.

Crossrefs

Cf. A051127, A106611 (number of distinct terms in each row), A374195 (row sums).

Programs

Mathematica

Table[Mod[Range[0, 10*(n-1), 10], n], {n, 15}]

Python

def A373917(n,k): return(k*10%n) # John Tyler Rascoe, Jul 02 2024