The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 26 13:56 EDT 2024. Contains 372826 sequences. (Running on oeis4.)