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A212558
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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.
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0
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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
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OFFSET
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10,2
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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sign,base,easy
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AUTHOR
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STATUS
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approved
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