A117800 -id:A117800 - cp's OEIS frontend

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

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

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

			T_2 enters a cycle of length 81 after 1 step.

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

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.

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

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}

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

A117831 Let S_n be the infinite sequence formed by starting with n and repeatedly reversing the digits and adding 4 to get the next term. Sequence gives number of steps for S_n to reach a cycle, or -1 if no cycle is ever reached.

Original entry on oeis.org

1, 1, 40, 7, 0, 0, 39, 6, 0, 0, 38, 5, 0, 18, 37, 3, 0, 43, 10, 0, 4, 42, 9, 4, 4, 41, 7, 0, 47, 40, 0, 8, 46, 13, 0, 8, 45, 11, 0, 7, 44, 0, 12, 50, 17, 3, 12, 49, 15, 1, 11, 48, 1, 16, 36, 3, 0, 16, 35, 1, 0, 41, 8, 2, 2, 40, 7, 2, 2, 39, 5, 0, 45, 12, 0, 6, 44, 11, 0, 6, 43, 9, 0, 49, 42, 0, 10

Offset: 1

N. J. A. Sloane, following discussions with Luc Stevens, May 03 2006

It is conjectured that S_n always reaches a cycle.

There are 22 different cycles of length 90 with 4-digit components. I guess that at most half of the numbers between 1000 and 10000 lead to the cycle of length 54 shown in A117830. - Klaus Brockhaus, May 05 2006

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

S_1 is given in A117828, S_3 in A117829, S_1015 in A117807.
Records are in A118473, A118474.
Full list of sequences on this topic (1): A117230, A117521, A117800, A117816, A117817, A117827, A117828, A117829, A117830, A117831 (this sequence)
Full list of sequences on this topic (2): A117837, A117841, A118473, A118474, A118510, A118511, A118512, A118513, A118514, A118515, A118516
Full list of sequences on this topic (3): A118517-A118533, A118535

V:= Vector(10^5,-1):
f:= proc(n)
  local L, H, S, i, j,found,x,y;
  global V;
  S:= {n}: H:= n; x:= n;
  for i from 1 to 10^5 do
    if V[x] > -1 then
       for j from 1 to i-1 do V[H[j]]:= i-j+V[x] od;
       return V[n];
    fi;
    L:= convert(x,base,10);
    x:= add(L[-j]*10^(j-1),j=1..nops(L)) + 4;
    if member(x, S) then
      found:= false; y:= 0;
      V[x]:= 0;
      for j from i by -1 to 1 do
        if H[j] = x then found:= true
        elif not found then V[H[j]]:= 0
        else y:= y+1; V[H[j]]:= y;
        fi
      od;
      return V[n]
    fi;
    H:= H, x;
    S:= S union {x};
  od;
end proc:
map(f, [$1..200]); # Robert Israel, May 07 2020

Corrected and extended by Klaus Brockhaus, May 05 2006

Confirmed by N. J. A. Sloane, May 05 2006

A117230 Start with 1 and repeatedly reverse the digits and add 1 to get the next term.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10

Offset: 1

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

N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A117830, A117521, A117828, A117800, A117816, A117817.

read transforms; t1:=[1]; for n from 1 to 80 do t1:=[op(t1),1+digrev(t1[n])]; od:

Join[{1},LinearRecurrence[{0,0,0,0,0,0,0,0,1},{2,3,4,5,6,7,8,9,10},99]] (* Ray Chandler, Jul 18 2015 *)

a(n)=if(n>1,(n-2)%9+2,1) \\ M. F. Hasler, May 22 2014

Has period 9.

G.f.: -x*(9*x^9 +9*x^8 +8*x^7 +7*x^6 +6*x^5 +5*x^4 +4*x^3 +3*x^2 +2*x +1) / ((x -1)*(x^2 +x +1)*(x^6 +x^3 +1)). - Colin Barker, May 23 2014

Correction to the terms of the sequence (inserted missing term) Jeremy Gardiner, Jun 17 2010

A118878 Let T(S,Q) be the sequence obtaining by starting with S and repeatedly reversing the digits and adding Q to get the next term. a(n) is the number of steps for T(n,5) to reach a cycle, or -1 if no cycle is ever reached.

Original entry on oeis.org

1, 190, 1, 87, 1, 0, 189, 0, 86, 0, 0, 188, 0, 85, 191, 0, 0, 95, 0, 190, 1, 0, 184, 0, 3, 187, 0, 0, 91, 0, 96, 0, 0, 180, 88, 0, 183, 0, 0, 87, 1, 92, 0, 0, 3, 84, 0, 179, 0, 0, 191, 0, 88, 0, 2, 190, 2, 87, 2, 1, 2, 97, 2, 192, 3, 0, 186, 0, 83, 189, 0, 0, 93, 0, 98, 0, 0, 182, 0, 1, 185, 0, 0, 89, 3, 94, 0, 0, 178, 86, 0, 181, 0, 0, 193, 0, 90, 0, 0, 1

Offset: 1

N. J. A. Sloane, May 26 2006

Is a(n) ever -1? If so then n > 10000.

The cycle has length 207 for S <= 1015, but for S = 1016 it has length 36 (see A118879).

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

Cf. A117828, A117800, A118879.

A118879 Let T(S,Q) be the sequence obtaining by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(1016,5), the first S for which T(S,5) reaches a cycle of length 36.

Original entry on oeis.org

1016, 6106, 6021, 1211, 1126, 6216, 6131, 1321, 1236, 6326, 6241, 1431, 1346, 6436, 6351, 1541, 1456, 6546, 6461, 1651, 1566, 6656, 6571, 1761, 1676, 6766, 6681, 1871, 1786, 6876, 6791, 1981, 1896, 6986, 6901, 1101, 1016, 6106, 6021

Offset: 0

N. J. A. Sloane, May 26 2006

The cycle is simply the first 36 terms, which then repeat.

For S < 1016, T(S,5) reaches a cycle of length 207 (cf. A117800).

N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A117828, A117800, A118878.

NestList[IntegerReverse[#]+5&,1016,40] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 05 2020 *)

A119903 Records in A118878.

Original entry on oeis.org

1, 190, 191, 192, 193, 8486, 16525, 16526, 16527, 16528, 16529, 16530, 825992, 1624591, 1624787, 1624983, 1625179, 1625375, 1625571, 1625767, 1625963, 1626159, 1626355, 1626391, 1626392, 1626393, 1626394, 1626395, 1626396, 82253978, 162059977, 162079973

Offset: 1

Klaus Brockhaus, May 28 2006

Rémy Sigrist, Table of n, a(n) for n = 1..56
Rémy Sigrist, C program
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

Cf. A117800, A118878, A119904.

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More terms from Rémy Sigrist, Aug 13 2022

A119904 Where records occur in A118878.

Original entry on oeis.org

1, 2, 15, 64, 95, 111, 119, 411, 604, 995, 1035, 4945, 10011, 10109, 10119, 10129, 10139, 10149, 10159, 10169, 10179, 10189, 10199, 11009, 40011, 60004, 99995, 100035, 499945, 1000011, 1001009, 1001019, 1001029, 1001039, 1001049, 1001059, 1001069, 1001079

Offset: 1

Klaus Brockhaus, May 28 2006

Rémy Sigrist, Table of n, a(n) for n = 1..56
Rémy Sigrist, C program
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

Cf. A117800, A118878, A119903.

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More terms from Rémy Sigrist, Aug 13 2022

A119902 Start with 100016 and repeatedly reverse the digits and add 5 to get the next term.

Original entry on oeis.org

100016, 610006, 600021, 120011, 110026, 620016, 610031, 130021, 120036, 630026, 620041, 140031, 130046, 640036, 630051, 150041, 140056, 650046, 640061, 160051, 150066, 660056, 650071, 170061, 160076, 670066, 660081, 180071, 170086, 680076

Offset: 1

Klaus Brockhaus, May 28 2006

Let T(S,Q) be the sequence obtained by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(10016,5). 100016 is the first S for which T(S,5) reaches a cycle of length 756. The cycle is simply the first 756 terms, which then repeat. A full period is given in the table.

Klaus Brockhaus, Table of n, a(n) for n = 1,...,756
N. J. A. Sloane and others, Sequences of RADD type, OEIS wiki.

Cf. A117800, A118879, A118878.

NestList[FromDigits[Reverse[IntegerDigits[#]]]+5&,100016,40]  (* Harvey P. Dale, Feb 24 2011 *)

cp's OEIS Frontend

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).

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A117831 Let S_n be the infinite sequence formed by starting with n and repeatedly reversing the digits and adding 4 to get the next term. Sequence gives number of steps for S_n to reach a cycle, or -1 if no cycle is ever reached.

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A117230 Start with 1 and repeatedly reverse the digits and add 1 to get the next term.

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A118878 Let T(S,Q) be the sequence obtaining by starting with S and repeatedly reversing the digits and adding Q to get the next term. a(n) is the number of steps for T(n,5) to reach a cycle, or -1 if no cycle is ever reached.

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A118879 Let T(S,Q) be the sequence obtaining by starting with S and repeatedly reversing the digits and adding Q to get the next term. This is T(1016,5), the first S for which T(S,5) reaches a cycle of length 36.

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Mathematica

A119903 Records in A118878.

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C

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A119904 Where records occur in A118878.

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C

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A119902 Start with 100016 and repeatedly reverse the digits and add 5 to get the next term.

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Mathematica