cp's OEIS Frontend

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

%I A276981 #14 Mar 25 2022 14:05:32
%S A276981 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,
%T A276981 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,
%U A276981 13,11,8,12,1,10,1,10,5,12,6,3,11,15,17,18,9,14,7,13,16,8,4,2
%N A276981 Irregular triangle T(n,k) read by rows of residue classes of powers of 10 modulo n.
%C A276981 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.
%C A276981 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, ...
%H A276981 Alois P. Heinz, <a href="/A276981/b276981.txt">Rows n = 1..800, flattened</a>
%e A276981 T(n,k), 1 <= k <= A051628(n) + A007732(n), starts with
%e A276981 n = 1:  0
%e A276981 n = 2:  1, 0
%e A276981 n = 3:  1
%e A276981 n = 4:  1, 2, 0
%e A276981 n = 5:  1, 0
%e A276981 n = 6:  1, 4
%e A276981 n = 7:  1, 3, 2, 6, 4, 5
%e A276981 n = 8:  1, 2, 4, 0
%e A276981 n = 9:  1
%e A276981 n = 10: 1, 0
%e A276981 n = 11: 1, 10
%e A276981 n = 12: 1, 10, 4
%e A276981 etc.
%p A276981 a:=proc(n)
%p A276981   local R,N,P,i;
%p A276981   R:=[seq(10^k mod n,k=0..n)]; # residue class
%p A276981   N:=[]; # nonperiodic part
%p A276981   P:=[]; # periodic part
%p A276981   for i from 1 to nops(R) do
%p A276981     member(R[i],R,'m');
%p A276981     if m<i then
%p A276981       if R[i]=1 then
%p A276981         P:=R[1..i-1];
%p A276981       else
%p A276981         N:=R[1..m-1];
%p A276981         P:=R[m..i-1];
%p A276981       fi;
%p A276981       break;
%p A276981     fi;
%p A276981   od;
%p A276981   return(op(N),op(P));
%p A276981 end:
%p A276981 seq(a(n),n=1..19);
%Y A276981 Cf. A007732, A051628.
%K A276981 nonn,base,tabf
%O A276981 1,6
%A A276981 _Martin Renner_, Apr 11 2017