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A039943 Every integer eventually goes to one of these under the "x goes to sum of squares of digits of x" map. 21
0, 1, 4, 16, 20, 37, 42, 58, 89, 145 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..9.

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. 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

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Last modified September 20 08:13 EDT 2019. Contains 327214 sequences. (Running on oeis4.)