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A336744
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Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly three.
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2
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14, 66, 94, 300, 384, 436, 496, 750, 1406, 1794, 2336, 2624, 28034
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OFFSET
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1,1
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COMMENTS
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Let b > 1 be an integer, and write the base b expansion of any nonnegative integer m as m = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.
Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(m) := x_0^2+ ... + x_d^2.
This is the 'sum of the squares of the digits' dynamical system alluded to in the name of the sequence.
It is known that the orbit set {m,S_{x^2,b}(m), S_{x^2,b}(S_{x^2,b}(m)), ...} is finite for all m>0. Each orbit contains a finite cycle, and for a given base b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,3).
A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. It is natural to wonder whether the set L(x^2,3) is infinite. It is a folklore conjecture that L(x^2,1) = {2,4}.
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LINKS
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D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.
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FORMULA
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EXAMPLE
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For instance, in base 14, the three cycles are (1), (37,85), and (25,122,164,221,123,185,178,244,46). To verify that (37,85) is a cycle in base 14, note that 37=9+2*14, and that 9^2+2^2=85. Similarly, 85=1+6*14, and 1^2+6^2=37.
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CROSSREFS
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Cf. A193583, A193585 (where cycles and fixed points are treated separately).
Cf. A336783 (4 cycles with sum of cubes of the digits).
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KEYWORD
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nonn,base,hard,more
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AUTHOR
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STATUS
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approved
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