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%I #29 Jun 10 2018 04:22:30
%S 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,
%T 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,
%U 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
%N Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.
%C Length of transient when the f[n]=Sum[digit^2 of n] function is iterated.
%C 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.
%D Hugo Steinhaus: "Sto zadan" (1958), "One Hundred Problems in Elementary Mathematics" (1964), problem 2. [From _M. F. Hasler_, May 24 2009]
%H T. D. Noe, <a href="/A099645/b099645.txt">Table of n, a(n) for n = 1..1000</a>
%H Arthur Porges, <a href="http://www.jstor.org/stable/2304639">A set of eight numbers</a>, Amer. Math. Monthly 52 (1945), 379-382. [_M. F. Hasler_, May 24 2009]
%e n=99999999999: iteration-list={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.
%t 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}];
%o (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
%o (Haskell)
%o a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
%o a099645_list = map a099645 [1..]
%o -- _Reinhard Zumkeller_, Aug 24 2011
%Y Cf. A007700, A031176.
%Y Cf. A000216, A003132, A003621
%Y 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
%K base,nonn
%O 1,3
%A _Labos Elemer_, Nov 08 2004
%E 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
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