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

4

`%I #13 Sep 14 2013 17:30:40
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`%S 366,680,691,1026,1136,1298,1323,1417,1464,1583,1604,1702,2079,2125,
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`%T 2222,2223,2374,2507,2604,2627,2821,2844,2897,3152,3157,3159,3183,
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`%U 3210,3231,3459,3697,3715,3762,3802,3866,3888,3936,3948,4004,4111,4133,4145,4231,4299
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`%N Numbers which do not reach zero under the repeated iteration x -> ceiling(sqrt(x)) * (ceiling(sqrt(x))^2 - x).
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`%C Ceiling equivalent of A219303, with somewhat different behavior despite a near-identical iterative process.
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`%C 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.
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`%C Conjecture #2: There are an infinite number of such finite loops, though there is often significant distance between them.
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`%C Conjecture #3: There are an infinite number of pairs of consecutive integers in this sequence despite being less abundant than in A219303.
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`%H Carl R. White, <a href="/A219960/b219960.txt">Table of n, a(n) for n = 1..10000</a>
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`%e 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.
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`%e 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.
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`%t 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 *)
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`%Y Cf. A219303, A219961, A219962, A219963
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`%K nonn
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`%O 1,1
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`%A _Carl R. White_, Dec 02 2012
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