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A219303 Numbers which do not reach zero under the repeated iteration x -> floor(sqrt(x)) * (x - floor(sqrt(x))^2).

%I A219303 #15 Sep 14 2013 17:29:00
%S A219303 8,14,15,18,23,28,39,51,53,65,83,139,172,174,194,197,198,226,292,318,
%T A219303 325,335,355,358,388,493,513,518,523,530,558,573,612,638,667,708,709,
%U A219303 715,717,748,761,766,785,830,832,847,858,863,864,935,944,979,999
%N A219303 Numbers which do not reach zero under the repeated iteration x -> floor(sqrt(x)) * (x - floor(sqrt(x))^2).
%C A219303 Collatz-like hailstone behavior is evident, but the iteration more closely resembles the iteration used to determine happy numbers (A007770), albeit in a non-base-specific manner. Unlike happy numbers, and despite being approximately as dense in the integers, these numbers do _not_ reach their iterative goal.
%C A219303 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.
%C A219303 Conjecture #2: There are an infinite number of such finite loops, though there is often significant distance between them.
%C A219303 Conjecture #3: There are an infinite number of pairs of consecutive integers in this sequence, e.g. 14 and 15, 197 and 198. This argument is strengthened by the fact there are other groupings such as triples - The first of these is 11527, 11528 and 11529 - and also that for randomly chosen numbers of hundreds of digits, N, the nearest pair or grouping appears to be within N +/- 1000.
%H A219303 Carl R. White, <a href="/A219303/b219303.txt">Table of n, a(n) for n = 1..10000</a>
%H A219303 Carl R. White and others, <a href="https://groups.google.com/forum/?fromgroups=#!msg/sci.math/3FbFc3FLALA/TpgOGBAQzdIJ">a posting in Usenet group sci.math</a>
%e A219303 355 is in the list because the iteration runs: 355 -> 558 -> 667 -> 1050 -> 832 -> 1344 -> 1728 -> [1927] -> 3354 -> 5985 -> 4312 -> 5655 -> 2250 -> [1927] -> etc. looping forever back to 1927.
%e A219303 99 is _not_ in the list because its iteration runs: 99 -> 162 -> 216 -> 280 -> 384 -> 437 -> 740 -> 297 -> 136 -> 165 -> 252 -> 405 -> 100 -> [0]
%e A219303 Note, however, that both chains exhibit hailstone behavior.
%o A219303 (PARI) i(n)=my(s=sqrtint(n));s*(n-s^2)
%o A219303 is(n)=my(m=i(n));while(m>7 && m!= n, n=i(n); m=i(i(m))); m>7 \\ termination subject to Conjecture 1
%o A219303 \\ _Charles R Greathouse IV_, Nov 30 2012
%K A219303 nonn
%O A219303 1,1
%A A219303 _Carl R. White_, Nov 17 2012
%E A219303 Conjectures and b-file added by _Carl R. White_, Nov 22 2012