%I
%S 0,4,6,12,4,18,5,2,0,19,0,12,2,6,24,12,14,0,16,12,6,19,7,18,25,28,
%T 9,2,18,6,8,20,0,30,44,0,4,4,36,28,35,21,3,19,0,30,19,12,36,35,
%U 24,26,6,36,8,24,6,8,18,24,35,15,9,46,16,45,7,28,45
%N a(n) = ((n  s)^2 mod (n + s))  ((n + s)^2 mod (n  s)), where s is the sum of the decimal digits of n.
%C Thus far (up to n=69) a pattern does not seems to exist, but we found something interesting: for every Poulet number not divisible by 3 checked, we obtained N=0 or N = ((n  s)^2 mod (n + s)).
%C Poulet numbers which give N=0: 341, 1105, 1729, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4681, 5461, 6601, 7957, 8321, 8911, 10261, 10585, 13741, 13747, 13981, 14491, 15709.
%C Poulet numbers which give N = ((n  s)^2 mod (n + s)): 1387.
%C We found out something even more: taking randomly a number n >= 10, that is 19, we obtained, taking beside s a number bigger than 0 and less than s, next values for N: 0, 0, 10, 14, 2, 18, 10, 10, 12.
%C For Poulet numbers checked (all the ones above) we got for N just the value 0 when we took, beside s, a number less than s. Still for Poulet numbers, we obtained a lot of values of 0 even when we took, beside s, a number bigger than s (but for 1105 we obtained, taking beside s the numbers 17, 18, 19, 20, the following values for N: 34, 36, 38, 40).
%C For n>2999 a(n) = 0, see proof in MoHPC link.  _Gerald Hillier_, Dec 18 2017
%H MoHPC: The Museum of HP Calculators, <a href="http://www.hpmuseum.org/forum/thread9719.html">Proof of Unproven Conjecture?</a>, December 2017.
%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%t Table[Mod[(n  s[n])^2, n + s[n]]  Mod[(n + s[n])^2, n  s[n]], {n, 10, 100}] (* _T. D. Noe_, May 21 2012 *)
%o (PARI) a(n)=my(s=sumdigits(n)); (ns)^2%(n+s)  (n+s)^2%(ns) \\ _Charles R Greathouse IV_, Dec 08 2014
%K sign,base,easy
%O 10,2
%A _Marius Coman_, May 21 2012
