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, -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, 9, 2, 18, 6, -8, -20, 0, 30, 44, 0, 4, 4, 36, 28, 35, 21, 3, -19, 0, -30, -19, -12, 36, 35, -24, 26, 6, 36, 8, -24, -6, 8, 18, 24, 35, 15, -9, 46, 16, 45, 7, 28, 45

Offset

10,2

Comments

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)).

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.

Poulet numbers which give N = -((n - s)^2 mod (n + s)): 1387.

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.

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).

For n>2999 a(n) = 0, see proof in MoHPC link. - Gerald Hillier, Dec 18 2017

Links

Table of n, a(n) for n=10..78.

MoHPC: The Museum of HP Calculators, Proof of Unproven Conjecture?, December 2017.

E. W. Weisstein, Poulet Number

Mathematica

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 *)

Prog

(PARI) a(n)=my(s=sumdigits(n)); (n-s)^2%(n+s) - (n+s)^2%(n-s) \\ Charles R Greathouse IV, Dec 08 2014

Crossrefs

Sequence in context: A116983 A196271 A078426 * A293836 A278252 A274298

Adjacent sequences: A212555 A212556 A212557 * A212559 A212560 A212561

Keyword

sign,base,easy

Author

Marius Coman, May 21 2012

Status

approved