OFFSET
0,3
COMMENTS
The subset of the first three terms also satisfies the current definition. An alternate definition would be: Periodic points of A003132. - M. F. Hasler, May 24 2009
Following I. Ja. Tanatar (Moscow), one can easily prove that, for a given x, there exists an iteration of the map f(x) given in the definition which reaches 1 or 89. Indeed, it is easy to see that if x has at least 3 digits, then f(x) < x. Therefore there exists an iteration of f with not more than 2 digits. For two-digit numbers the property is verified directly. See Kordemsky. - Vladimir Shevelev, May 06 2013
REFERENCES
B. A. Kordemsky, Matematicheskaja Smekalka, Moscow, 1955, pp. 305 and 557 (in Russian).
LINKS
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379-382. [Annotated scanned copy]
Eric W. Weisstein, MathWorld: Happy Number
Wikipedia, Periodic point
MATHEMATICA
lst = {}; Do[a = NestWhile[Plus @@ (IntegerDigits@#^2) &, n, Unequal, All]; If[FreeQ[lst, a], AppendTo[lst, a]], {n, 10^4}] (* Robert G. Wilson v, Jan 19 2006 *)
Union[Table[NestWhile[Total[IntegerDigits[#]^2]&, n, Unequal, All], {n, 0, 100}]] (* Harvey P. Dale, Nov 26 2013 *)
PROG
(Haskell)
a039943 n = a039943_list !! n
a039943_list = [0, 1, 4, 16, 20, 37, 42, 58, 89, 145]
-- Reinhard Zumkeller, Mar 16 2013
CROSSREFS
Cf. A003132 (the iterated map), A003621, A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4, the only nontrivial limit cycle), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169).
KEYWORD
nonn,fini,full,base
AUTHOR
STATUS
approved