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

1,3

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. - 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

Example

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.

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}]

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..]
-- Reinhard Zumkeller, Aug 24 2011

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

Sequence in context: A062521 A271951 A157700 * A220412 A199092 A167260

Adjacent sequences: A099642 A099643 A099644 * A099646 A099647 A099648

Keyword

base,nonn

Author

Labos Elemer, Nov 08 2004

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