A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

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

Offset: 1

Author

Labos Elemer, Nov 08 2004

Keywords

• base
• nonn

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.

Examples

			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.
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382. - _M. F. Hasler_, May 24 2009

Crossrefs

Cf. A007700, A031176.

Cf. A000216, A003132, A003621

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

Programs

Haskell

a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
a099645_list = map a099645 [1..]
-- Reinhard Zumkeller, Aug 24 2011

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]]]; Table[a[n], {n, 1, 256}]

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

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