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

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

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

Table of n, a(n) for n=1..26.

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

Adjacent sequences: A083512 A083513 A083514 * A083516 A083517 A083518

Keyword

easy,nonn

Author

Jason Earls, Jun 09 2003

Status

approved