A117816 Number of steps until the RADD sequence T(k+1) = n + R(T(k)), T(0) = 1, enters a cycle; -1 if no cycle is ever reached. (R=A004086: reverse digits)

1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 2, 31, 15, -1, 721, 9, 1, 6, -1, 3, 5, 28, 29, 131, 23, 1, 31, 6, -1, 1, 19, 1, 53, 4, 406, 34, 254, 8, -1, 3, 245, 1, 3, 2, 422, 42, 308, 1, -1, 2, 2, 49, 1, 1371, 13, 1, 1, 2, -1, 78, 65, 1, 809, 1575, 5, 43, 31, 2, -1, 33, 2, 21, 192, 857, 91, 1, 2, 2, -1, 2, 491, 1, 2, 1, 81, 49, 1, 2, -1, 35, 197, 72, 1, 12, 79, 1, 6004, 1, -1, 52, 10264, 9, 28, 2, 2, 1, 427, 1, -1, 1, 1, 49, 167

Offset

1,12

Comments

Comments following discussions with David Applegate, May 05 2006: (Start)

Certainly a(10) = -1 and probably a(n) is always -1 if n is a multiple of 10. Furthermore a(15) is almost certainly -1: T_15 has not reached a cycle in 10^7 terms (see A118532).

(End)

If n is a multiple of 10 the operation can never generate a trailing zero and so is reversible. So it loops only if it returns to the start, which is impossible. Hence a(10k) = -1. - Martin Fuller, May 12 2006

I suspect a(115) = 385592406, A117817(115) = 79560. Can someone confirm? - Martin Fuller, May 12 2006

The map f: x -> R(x)+n is injective, f(x)=f(y) <=> R(x)=R(y) <=> x=y, unless x or y only differ in trailing zeros. For n=10k, however, trailing zeros can never occur. (This also implies that the terms are of increasing length.) Thus, for n=10k, no number can occur twice in the orbit of 1 under f, i.e., a(10k)=-1. A sketch of proof for a(15)=-1 is given in A118532. As of today, no other n with a(n)=-1 seems to be known. - M. F. Hasler, May 06 2012

Links

Table of n, a(n) for n=1..114.

N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

Example

T_2 enters a cycle of length 81 after 1 step.

Mathematica

ReverseNum[n_] := FromDigits[Reverse[IntegerDigits[n]]]; maxLen=10000; Table[z=1; lst={1}; While[z=ReverseNum[z]+n; !MemberQ[lst, z] && Length[lst]<maxLen, AppendTo[lst, z]]; If[Length[lst]<maxLen, Position[lst, z][[1, 1]]-1, -1], {n, 100}] (* T. D. Noe *)

Prog

(PARI) A117816(n, L=10^5, S=1)={ for(F=0, 1, my(u=Vecsmall(S)); while(L-- & #u<#u=vecsort(concat(u, Vecsmall(S=A004086(S)+n)), , 8), ); L || F=1; /* 1st run counts until repetition, now subtract cycle length */ F || L=1+#u); L-1}

Crossrefs

For T_1, T_2, ..., T_16 (omitting T_9, which is uninteresting) see A117230, A117521, A118517, A117828, A117800, A118525, A118526, A118527, A117841, A118528, A118529, A118530, A118531, A118532, A118533.

Cf. A117817.

Sequence in context: A178194 A069460 A352959 * A237263 A099189 A247099

Adjacent sequences: A117813 A117814 A117815 * A117817 A117818 A117819

Keyword

sign,base

Author

N. J. A. Sloane, following discussions with Luc Stevens, May 04 2006

Extensions

a(21)-a(33) from Luc Stevens, May 08 2006

a(33) onwards from T. D. Noe, May 10 2006

Further terms from Martin Fuller, May 12 2006

Status

approved