

A039943


Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map.


21




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 twodigit 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

Table of n, a(n) for n=0..9.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379382.
Arthur Porges, A set of eight numbers, Amer. Math. Monthly, 52 (1945), 379382. [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. A000216, A003621.
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).
Sequence in context: A071966 A326781 A326788 * A193996 A232400 A067671
Adjacent sequences: A039940 A039941 A039942 * A039944 A039945 A039946


KEYWORD

nonn,fini,full,base


AUTHOR

N. J. A. Sloane


STATUS

approved



