A test for Belgian-ity

[Paul C. Leopardi]

Hi Eugene,
Reply below.
Best regards, Paul Leopardi
On Thu, 16 Jun 2005 05:35 am, Eugene McDonnell wrote:
> More or less empirically, I have found that I can make an easy test to
> see if a given positive integer is Belgian; I'd appreciate it if
> someone more versed in mathematics could prove or disprove the
> following, before I pollute the Encyclopedia with an incorrect comment
> for A106039:
>
> To test the Belgian-ity of a positive integer n, find s, the cumulative
> sum of its digits, and m the sum of its digits; then n is Belgian if

 From the example below, I think you mean that s is the sequence of cumulative 
sums of the digits of n. At least this wording seems clearer to me.

> there is a 0 in any of the differences of n minus the items of s, mod

Do you mean that one of the elements of the sequence is zero, or that one of 
the elements of the sequence contains the digit '0' ? Your more exotic 
example says that it is the first case, since there is a '0' in 20, but you 
did not mention this.

> m. For example, if n is 176 then s is 1 8 14, and m is 14; 176 - 1 8 14
> is 175 168 162, and the residues mod(14) are 7 0 8, so 176 is Belgian.
> Contrariwise, if n is 177, s is 1 8 15, and m is 15; 177 - 1 8 15 is
> 176 169 162, and the residues, mod(15) are 11 4 12, so 177 is not
> Belgian.
>
> A more exotic example:
>
> Let n be 1234567898765; then s is 1 3 6 10 15 21 28 36 45 53 60 66 71,
> and m is 71.
>
>     (1234567898765 - 1 3 6 10 15 21 28 36 45 53 60 66 71) mod(71) is:
>
> 20 18 15 11 6 0 64 56 47 39 32 26 21
>
> which contains a 0, so n is Belgian.
>
> Eugene McDonnell