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A083515 Smallest number k such that R(n) + k is a square, where R(n) = A002275. 0

%I #11 Jun 23 2019 02:26:05

%S 0,5,10,45,125,445,1914,4445,1570,44445,156989,444445,941538,4444445,

%T 9826365,44444445,139362425,444444445,1287131805,4444444445,

%U 2222074045,44444444445,11388893810,444444444445,1138889380989,4444444444445

%N Smallest number k such that R(n) + k is a square, where R(n) = A002275.

%C For n > 1, a(n) is never zero. Proof: R(2) = 4*2+3, R(3) = 4*27+3. In general 111...1 = 4*2777...7+3, or if f(n) = 2777...7 = (7/9)*(-1 + 10^n) + 2^(n+1)*5^n, then R(n) = 4*f(n-2) + 3. Hence repunits are of the form 4m+3 and cannot be square. [Paraphrased from Tattersall]

%D J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001. pp. 57, 330.

%F a(n) = ceiling(sqrt(R(n)))^2 - R(n).

%e a(5) = 125 because 11111 + 125 = 11236 is a square.

%Y Cf. A002275.

%K easy,nonn

%O 1,2

%A _Jason Earls_, Jun 09 2003

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