

A099645


Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.


16



0, 1, 5, 0, 4, 9, 5, 5, 4, 1, 2, 5, 2, 6, 3, 0, 5, 3, 4, 0, 5, 6, 3, 1, 3, 2, 6, 3, 2, 5, 2, 3, 4, 4, 5, 8, 0, 2, 5, 1, 6, 0, 4, 4, 7, 4, 3, 6, 4, 4, 3, 3, 5, 7, 5, 2, 4, 0, 2, 9, 1, 2, 8, 4, 2, 7, 2, 2, 5, 5, 5, 6, 1, 3, 4, 2, 2, 4, 3, 5, 3, 3, 2, 6, 1, 2, 4, 7, 0, 4, 4, 2, 5, 4, 2, 5, 3, 1, 8, 1, 2, 5, 2, 6, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Length of transient when the f[n]=Sum[digit^2 of n] function is iterated.
In A031176 including cycle lengths[=c] of this iteration only c=1 and c=8 occur. A007770 lists cases of c=1, the happy numbers.


REFERENCES

Hugo Steinhaus: "Sto zadan" (1958), "One Hundred Problems in Elementary Mathematics" (1964), problem 2. [From M. F. Hasler, May 24 2009]


LINKS



EXAMPLE

n=99999999999: iterationlist={99999999999,891,146,53,34,25,29,85,89,145,42,20,[4,16,37,58,89,145,42,20],4,...]}. Lengths of transient=12, of cycle=8.


MATHEMATICA

fu[x_] :=Apply[Plus, IntegerDigits[x]^2]; hs=20; transient lengths are obtained by: a[n_] :=1+Min[Flatten[Position[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]] Last[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]]], 0]]], {n, 1, 256}];


PROG

(PARI) A099645(n)={ local( c=0, S=Set([1, 4, 16, 37, 58, 89, 145, 42, 20])); while( !setsearch(S, n), n=A003132(n); c++); c} \\ M. F. Hasler, May 24 2009
(Haskell)
a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
a099645_list = map a099645 [1..]


CROSSREFS

Cf. A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169).  M. F. Hasler, May 24 2009


KEYWORD

base,nonn


AUTHOR



EXTENSIONS

Terms checked using the given PARI code. However, according to the domain of A003132 and the definition of A039943 (which both include 0), an initial a(0)=0 should be added here, too.  M. F. Hasler, May 24 2009


STATUS

approved



