A276981 Irregular triangle T(n,k) read by rows of residue classes of powers of 10 modulo n.

0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 4, 1, 3, 2, 6, 4, 5, 1, 2, 4, 0, 1, 1, 0, 1, 10, 1, 10, 4, 1, 10, 9, 12, 3, 4, 1, 10, 2, 6, 4, 12, 8, 1, 10, 1, 10, 4, 8, 0, 1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12, 1, 10, 1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2

Offset

1,6

Comments

The length of the nonperiodic part of the residue class values is given in A051628, the length of the periodic part is given in A007732.

These residue class values are useful to check the divisibility of a number by the divisor n simply by calculating the weighted sum of digits. For example, the number 86415 is divisible by 7, because the weighted sum of digits 5*1 + 1*3 + 4*2 + 6*6 + 8*4 = 84 is divisible by 7. The used weights are the residue class values for n = 7: 1, 3, 2, 6, 4, 5, ... for ones, tens, hundreds, ...

Links

Alois P. Heinz, Rows n = 1..800, flattened

Example

T(n,k), 1 <= k <= A051628(n) + A007732(n), starts with
n = 1:  0
n = 2:  1, 0
n = 3:  1
n = 4:  1, 2, 0
n = 5:  1, 0
n = 6:  1, 4
n = 7:  1, 3, 2, 6, 4, 5
n = 8:  1, 2, 4, 0
n = 9:  1
n = 10: 1, 0
n = 11: 1, 10
n = 12: 1, 10, 4
etc.

Maple

a:=proc(n)
  local R, N, P, i;
  R:=[seq(10^k mod n, k=0..n)]; # residue class
  N:=[]; # nonperiodic part
  P:=[]; # periodic part
  for i from 1 to nops(R) do
    member(R[i], R, 'm');
    if m<i then
      if R[i]=1 then
        P:=R[1..i-1];
      else
        N:=R[1..m-1];
        P:=R[m..i-1];
      fi;
      break;
    fi;
  od;
  return(op(N), op(P));
end:
seq(a(n), n=1..19);

Crossrefs

Cf. A007732, A051628.

Sequence in context: A364955 A112517 A112519 * A230305 A357293 A357119

Adjacent sequences: A276978 A276979 A276980 * A276982 A276983 A276984

Keyword

nonn,base,tabf

Author

Martin Renner, Apr 11 2017

Status

approved