A051127 -id:A051127 - cp's OEIS frontend

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

Paolo Xausa, Jun 26 2024

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).

			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.

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

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

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

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

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 3, 4, 1, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 1, 0, 0, 4, 3, 0, 1, 0, 1, 2, 1, 1, 1, 4, 1, 8, 1, 0, 0, 0, 0, 2, 0, 5, 0, 0, 4, 1, 0, 1, 1, 1, 3, 1, 6, 1, 1, 9, 2, 1, 0, 0, 2, 0, 4, 4, 0, 0, 8, 6, 3, 4, 1, 0, 1, 0, 1, 0, 3, 1, 1, 0, 5, 4, 9, 2, 1

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A324471 a(n) = 10 mod n.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Mar 07 2019, following a suggestion from Charles Kusniec

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010692, A010879, A048158, A051127, A090976, A324472.

Mod[10,Range[100]] (* Paolo Xausa, Nov 14 2023 *)

From Elmo R. Oliveira, Aug 03 2024: (Start)
G.f.: x^3*(1 + x - 2*x^2 + 4*x^3 - x^4 - x^5 - x^6 - x^7 + 10*x^8)/(1 - x).
a(n) = 10 for n > 10. (End)

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

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A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

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A324471 a(n) = 10 mod n.

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