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A212558 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. 0

%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/thread-9719.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)); (n-s)^2%(n+s) - (n+s)^2%(n-s) \\ _Charles R Greathouse IV_, Dec 08 2014

%K sign,base,easy

%O 10,2

%A _Marius Coman_, May 21 2012

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Last modified March 26 00:16 EDT 2019. Contains 321477 sequences. (Running on oeis4.)