# [seqfan] Scott Shannon's "Grow" sequence A332580 (and A332584)

Neil Sloane njasloane at gmail.com
Tue Feb 18 16:27:28 CET 2020

```The definition of A332580  is:

a(n) = minimal positive k such that the concatenation of the decimal digits
of n,n+1,...,n+k is divisible by n+k+1, or -1 if no such k exists

.and A332584 gives the values of n+k.

Examples:

a(1) = 1 as '1' || '2' = '12', which is divisible by 3 (where || denotes
decimal concatenation).

a(7) = 13 as '7' || '8' || '9' || '10' || '11' || '12' ||  ... || '20' =
7891011121314151617181920, which is divisible by 21.

a(8) = 2 as '8' || '9' || '10' = 8910, which is divisible by 11.

a(2) = 80: the concatenation 2 || 3 || ... || 82 is

23456789101112131415161718192021222324252627282930313233343536373839\

40414243444546474849505152535455565758596061626364656667686970717273747\

576777879808182, which is divisible by 83.

What makes this interesting is that the chance that a particular k works
is, naively, 1/(n+k+1).  For fixed n, the sum 1/(n+k+1) diverges, so a(n)
should always exist. However, Scott and I have not been able to find three
values up to 100, namely a(44), a(92), and a(98).

Possibly there are number-theoretic reasons why 44, 92, 98, ... are
special, or maybe we did not search far enough.

I ran Maple out to 200000, on 44, but when I tried to go out to a million,
Maple managed to put my iMac into a coma and I had to turn off the power.
Scott ran a Java program a lot further, but I always worry when Java is
doing number theory.

So we need help! Find a(44), a(92), a(98) or show they do not exist.

```