cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A219552 First differences of A219303.

Original entry on oeis.org

6, 1, 3, 5, 5, 11, 12, 2, 12, 18, 56, 33, 2, 20, 3, 1, 28, 66, 26, 7, 10, 20, 3, 30, 105, 20, 5, 5, 7, 28, 15, 39, 26, 29, 41, 1, 6, 2, 31, 13, 5, 19, 45, 2, 15, 11, 5, 1, 71, 9, 35, 20, 13, 2, 36, 1, 15, 12, 17, 20, 14, 45, 4, 27, 12, 38, 30, 3, 25, 7, 24
Offset: 1

Views

Author

Carl R. White, Nov 22 2012

Keywords

Comments

Conjecture: 1 appears infinitely often in this sequence (indicating an infinite number of pairs of consecutive integers in the parent sequence).
Observation: Values tend to average around 20 for early terms, though some evidence suggests this average falls slightly much later. It is not clear how the overall average value behaves approaching the limit at infinity.

Links

Crossrefs

Cf. A219303, A219553.

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

Original entry on oeis.org

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

Views

Author

Carl R. White, Dec 02 2012

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A219303, A219961, A219962, A219963

Programs

  • 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 *)

A219553 Records in A219552.

Original entry on oeis.org

6, 11, 12, 18, 56, 66, 105, 110, 130, 132, 139, 140, 185, 195, 197, 202, 261, 266, 313
Offset: 1

Views

Author

Carl R. White, Nov 22 2012

Keywords

Comments

313 found at x = 79142421 under the iteration of A219303. No more entries up to x = 101890000. - Carl R. White, Nov 30 2012

Crossrefs

Cf. A219303, A219552, A219962.

Extensions

a(17)-a(19) from Carl R. White, Nov 30 2012

A219963 Numbers which do not reach zero under either of the iterations: x -> floor(sqrt(x)) * (x - floor(sqrt(x))^2) or y -> ceiling(sqrt(y)) * (ceiling(sqrt(y))^2 - y).

Original entry on oeis.org

2897, 3159, 3183, 4004, 6335, 7025, 8163, 8237, 8621, 9234, 12204, 12963, 13381, 14340, 15217, 16191, 16438, 17474, 17763, 17972, 18065, 18990, 19677, 19848, 20345, 20803, 21426, 21539, 22022, 25834, 26872, 27175, 28052, 28929, 28996, 29295, 30511, 30991
Offset: 1

Views

Author

Carl R. White, Dec 02 2012

Keywords

Comments

Intersection of A219303 and A219960.
Like the parent sequences, this sequence has pairs of consecutive integers; The first of these pairs is 89971 and 89972.
It is possible, assuming the infinite-pairs conjectures are true for both parent sequences, that there may also be an infinite number of pairs here, but even then that is not guaranteed.

Links

Crossrefs

Cf. A219303, A219960.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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