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A083515 Smallest number k such that R(n) + k is a square, where R(n) = A002275. 0
0, 5, 10, 45, 125, 445, 1914, 4445, 1570, 44445, 156989, 444445, 941538, 4444445, 9826365, 44444445, 139362425, 444444445, 1287131805, 4444444445, 2222074045, 44444444445, 11388893810, 444444444445, 1138889380989, 4444444444445 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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]
REFERENCES
J. Tattersall, "Elementary Number Theory in Nine Chapters". Cambridge University Press, 2001. pp. 57, 330.
LINKS
FORMULA
a(n) = ceiling(sqrt(R(n)))^2 - R(n).
EXAMPLE
a(5) = 125 because 11111 + 125 = 11236 is a square.
CROSSREFS
Cf. A002275.
Sequence in context: A187877 A328605 A122173 * A343467 A103971 A270085
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Jun 09 2003
STATUS
approved

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Last modified April 19 08:39 EDT 2024. Contains 371782 sequences. (Running on oeis4.)