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

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

Offset: 1

Author

Carl R. White, Nov 17 2012

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Carl R. White, Table of n, a(n) for n = 1..10000
• Carl R. White and others, a posting in Usenet group sci.math

Programs

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

Extensions

Conjectures and b-file added by Carl R. White, Nov 22 2012