A219960 Numbers which do not reach zero under the repeated iteration x -> ceiling(sqrt(x)) * (ceiling(sqrt(x))^2 - x).

366, 680, 691, 1026, 1136, 1298, 1323, 1417, 1464, 1583, 1604, 1702, 2079, 2125, 2222, 2223, 2374, 2507, 2604, 2627, 2821, 2844, 2897, 3152, 3157, 3159, 3183, 3210, 3231, 3459, 3697, 3715, 3762, 3802, 3866, 3888, 3936, 3948, 4004, 4111, 4133, 4145, 4231, 4299

Offset

1,1

Comments

Ceiling equivalent of A219303, with somewhat different behavior despite a near-identical iterative process.

Conjecture #1: All numbers under the iteration reach 0 or, like the elements of this sequence, reach a finite loop, and none expand indefinitely to infinity.

Conjecture #2: There are an infinite number of such finite loops, though there is often significant distance between them.

Conjecture #3: There are an infinite number of pairs of consecutive integers in this sequence despite being less abundant than in A219303.

Links

Carl R. White, Table of n, a(n) for n = 1..10000

Example

1702 is in this list as 38 iterations return to 1702. Many other numbers reach this loop. 5832 is also in this list and is the smallest member of a different loop.
1703 is _not_ in this list because the iteration runs: 1703 -> 2562 -> 1989 -> 1620 -> 2501 -> 5100 -> 6048 -> 2808 -> 53 -> 88 -> 120 -> 11 -> 20 -> 25 -> 0.

Mathematica

f[n_] := Ceiling[Sqrt[n]]*(Ceiling[Sqrt[n]]^2 - n); Select[Range[5000], NestWhileList[f, #, UnsameQ, All][[-1]] > 0 &] (* T. D. Noe, Dec 04 2012 *)

Crossrefs

Cf. A219303, A219961, A219962, A219963

Sequence in context: A073305 A248552 A259077 * A033174 A350471 A249704

Adjacent sequences: A219957 A219958 A219959 * A219961 A219962 A219963

Keyword

nonn

Author

Carl R. White, Dec 02 2012

Status

approved