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A219303
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Numbers which do not reach zero under the repeated iteration x -> floor(sqrt(x)) * (x - floor(sqrt(x))^2).
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5
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8, 14, 15, 18, 23, 28, 39, 51, 53, 65, 83, 139, 172, 174, 194, 197, 198, 226, 292, 318, 325, 335, 355, 358, 388, 493, 513, 518, 523, 530, 558, 573, 612, 638, 667, 708, 709, 715, 717, 748, 761, 766, 785, 830, 832, 847, 858, 863, 864, 935, 944, 979, 999
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OFFSET
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1,1
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COMMENTS
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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.
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, 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.
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LINKS
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Carl R. White, Table of n, a(n) for n = 1..10000
Carl R. White and others, a posting in Usenet group sci.math
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EXAMPLE
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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.
99 is _not_ in the list because its iteration runs: 99 -> 162 -> 216 -> 280 -> 384 -> 437 -> 740 -> 297 -> 136 -> 165 -> 252 -> 405 -> 100 -> [0]
Note, however, that both chains exhibit hailstone behavior.
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PROG
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(PARI) i(n)=my(s=sqrtint(n)); s*(n-s^2)
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
\\ Charles R Greathouse IV, Nov 30 2012
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CROSSREFS
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Sequence in context: A226799 A236295 A178838 * A082772 A103338 A250004
Adjacent sequences: A219300 A219301 A219302 * A219304 A219305 A219306
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KEYWORD
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nonn
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AUTHOR
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Carl R. White, Nov 17 2012
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EXTENSIONS
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Conjectures and b-file added by Carl R. White, Nov 22 2012
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STATUS
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approved
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