A373917 - OEIS

%I #39 Sep 09 2024 18:45:31

%S 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,

%T 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,

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

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

%C 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).

%H John Tyler Rascoe, <a href="/A373917/b373917.txt">Rows n = 1..150, flattened</a>

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=Ki-M1DJIZsk">Solving Seven</a>, Numberphile YouTube video, 2024.

%e Triangle begins:

%e   n\k| 0  1  2  3  4  5  6  7  8  9

%e   ---------------------------------

%e    1 | 0;

%e    2 | 0, 0;

%e    3 | 0, 1, 2;

%e    4 | 0, 2, 0, 2;

%e    5 | 0, 0, 0, 0, 0;

%e    6 | 0, 4, 2, 0, 4, 2;

%e    7 | 0, 3, 6, 2, 5, 1, 4;

%e    8 | 0, 2, 4, 6, 0, 2, 4, 6;

%e    9 | 0, 1, 2, 3, 4, 5, 6, 7, 8;

%e   10 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

%e   ...

%e Suppose m = 3714289 and you want to determine m mod 7 (the example shown in the video).

%e Start with the first digit of m (3) and calculate T(7,3 mod 7) = T(7,3) = 2.

%e Add it to the next digit of m (7) and calculate T(7,(2+7) mod 7) = T(7,2) = 6.

%e Add it to the next digit of m (1) and calculate T(7,(6+1) mod 7) = T(7,0) = 0.

%e Add it to the next digit of m (4) and calculate T(7,(0+4) mod 7) = T(7,4) = 5.

%e Add it to the next digit of m (2) and calculate T(7,(5+2) mod 7) = T(7,0) = 0.

%e Add it to the next digit of m (8) and calculate T(7,(0+8) mod 7) = T(7,1) = 3.

%e Add it to the final digit of m (9) and calculate (3+9) mod 7 = 5, which corresponds to 3714289 mod 7.

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

%o (Python)

%o def A373917(n,k): return(k*10%n) # _John Tyler Rascoe_, Jul 02 2024

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

%K nonn,tabl

%O 1,6

%A _Paolo Xausa_, Jun 26 2024