A004086 -id:A004086 - 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

A102111 Iccanobirt numbers (1 of 15): a(n) = a(n-1) + a(n-2) + R(a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 99, 185, 328, 612, 1521, 2956, 4693, 8900, 20185, 33049, 53332, 144483, 291848, 459666, 1135955, 2443813, 4246722, 12285846, 19716010, 34278280, 118852511, 154192582, 281332336, 550783729, 1117407516, 2301424427

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Cf. A102112-A102125, A102131, A102171.

a:=[0,0,1];[n le 3 select a[n] else Self(n-1) + Self(n-2) + Seqint(Reverse(Intseq(Self(n-3)))):n in [1..36]]; // Marius A. Burtea, Oct 23 2019

read("transforms") ;
A102111 := proc(n)
    option remember;
    if n <= 2 then
        return op(n+1,[0,0,1]) ;
    else
        return procname(n-1)+procname(n-2)+digrev(procname(n-3)) ;
    end if;
end proc:
seq(A102111(n),n=0..20) ; # R. J. Mathar, Nov 17 2012

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=a[n-1]+a[n-2]+R[a[n-3]];Table[a[n], {n, 0, 40}]
nxt[{a_,b_,c_}]:={b,c,IntegerReverse[a]+b+c}; NestList[nxt,{0,0,1},40][[;;,1]] (* Harvey P. Dale, Jul 18 2023 *)

def R(n):
  n_str = str(n)
  reversedn_str = n_str[::-1]
  reversedn = int(reversedn_str)
  return reversedn
def A(n):
  if n == 0:
    return 0
  elif n == 1:
    return 0
  elif n == 2:
    return 1
  elif n >= 3:
    return A(n-1)+A(n-2)+R(A(n-3))
for i in range(0,20):
  print(A(i)) # Dylan Delgado, Oct 23 2019

A004086(a(n)) = A102119(n).

A102125 Iccanobirt numbers (15 of 15): a(n) = R(R(a(n-1)) + R(a(n-2)) + R(a(n-3))), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 31, 42, 44, 18, 941, 472, 405, 729, 5071, 6313, 8675, 90601, 31591, 9853, 11733, 31865, 31149, 736481, 365533, 313416, 3154311, 9834802, 5123383, 7112507, 12796921, 35055832, 19867834, 56610708, 906334841, 561210372

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A102111-A102124.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, binomial(n,2),
      R(R(a(n-1)) + R(a(n-2)) + R(a(n-3))))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 18 2014

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[R[a[n-1]]+R[a[n-2]]+R[a[n-3]]];Table[a[n], {n, 0, 40}]
rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]]; nxt[{a_, b_, c_}] := {b, c, rev[rev[a] + rev[b] + rev[c]]}; Transpose[NestList[nxt,{0,0,1},40]][[1]] (* Harvey P. Dale, Mar 20 2015 *)
nxt[{a_,b_,c_}]:=With[{ir=IntegerReverse},{b,c,ir[ir[a]+ir[b]+ir[c]]}]; NestList[nxt,{0,0,1},40][[;;,1]] (* Harvey P. Dale, Jul 22 2025 *)

a(n) = A004086(A102117(n)).

A061205 a(n) = n times R(n) where R(n) (A004086) is the digit reversal of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 121, 252, 403, 574, 765, 976, 1207, 1458, 1729, 40, 252, 484, 736, 1008, 1300, 1612, 1944, 2296, 2668, 90, 403, 736, 1089, 1462, 1855, 2268, 2701, 3154, 3627, 160, 574, 1008, 1462, 1936, 2430, 2944, 3478, 4032, 4606

Offset: 0

Amarnath Murthy, Apr 21 2001

Every third term is divisible by 9, no other term is divisible by 3. - Alonso del Arte, Mar 04 2013

			a(10) = 10 = 10 * 01.
a(11) = 121 = 11 * 11.
a(12) = 252 = 12 * 21.
a(13) = 403 = 13 * 31.

Harry J. Smith and Indranil Ghosh, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)

Cf. A004086, A203924 (triple repetitions).

a061205 n = a004086 n * n
-- Reinhard Zumkeller, Apr 10 2012, Apr 29 2011

#*FromDigits[Reverse[IntegerDigits[#]]] &/@Range[0, 49]  (* Ant King, Jan 07 2012 *)
#*IntegerReverse[#]& /@ Range[0, 49] (* Jean-François Alcover, Oct 27 2019 *)

rev(k) = subst(Polrev(digits(k)), x, 10);
a(n) = n*rev(n); \\ Michel Marcus, Feb 14 2015

a(n) = n*fromdigits(Vecrev(digits(n))); \\ Michel Marcus, May 28 2018

def A061205(n):
    return n*A004086(n) # Indranil Ghosh, Jan 09 2017

Corrected and extended by Patrick De Geest, Jun 04 2001

A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 13, 41, 51, 61, 17, 81, 19, 2, 12, 22, 23, 42, 52, 26, 72, 82, 29, 3, 31, 23, 33, 241, 53, 63, 37, 281, 39, 4, 41, 24, 43, 44, 54, 46, 47, 84, 94, 5, 312, 421, 53, 45, 55, 65, 372, 481, 59, 6, 61, 62, 36, 46, 551, 66, 67, 482, 69, 7, 71, 27

Offset: 1

J. H. Conway & Tanya Khovanova, Jun 14 2009

Might be called TITO(n), turning n inside out then turning outside in.

Here is the operation: take a number n and find its prime factors. Reverse the digits of every prime factor (for example, replace 17 by 71). Multiply the factors respecting multiplicities. For example, if the original number was 17^2*43^3, the new product will be 71^2*34^3. After that, reverse the resulting number.

			a(34) = 241, because 34 = 2*17, f(34) = 2*71 = 142, and reversing gives 241.

M. F. Hasler, Table of n, a(n) for n=1..5000. [From _M. F. Hasler_, Jun 24 2009]
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A161597, A161598, A161600, A071786, A004086, A151764.

a161594 = a004086 . a071786  -- Reinhard Zumkeller, Oct 14 2011

read("transforms") ; A071786 := proc(n) local ifs,a,d ; ifs := ifactors(n)[2] ; a := 1 ; for d in ifs do a := a*digrev(op(1,d))^op(2,d) ; od: a ; end: A161594 := proc(n) digrev(A071786(n)) ; end: seq(A161594(n),n=1..80) ; # R. J. Mathar, Jun 16 2009
# second Maple program:
r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 19 2017

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Table[f[n], {n, 100}]
Table[IntegerReverse[Times@@Flatten[Table[IntegerReverse[#[[1]]],#[[2]]]& /@FactorInteger[n]]],{n,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 21 2016 *)

R=A004086; A161594(n)={n=factor(n);n[,1]=apply(R,n[,1]);R(factorback(n))} \\  M. F. Hasler, Jun 24 2009. Removed code for R here, see A004086 for most recent & efficient version. - M. F. Hasler, May 11 2015

from math import prod
from sympy import factorint
def f(n): return prod(int(str(p)[::-1])**e for p, e in factorint(n).items())
def R(n): return int(str(n)[::-1])
def a(n): return 1 if n == 1 else R(f(n))
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 28 2022

a(p) = p, for prime p.
a(A161598(n)) <> A161598(n); a(A161597(n)) = A161597(n); A010051(a(A161600(n))) = 1.
From M. F. Hasler, Jun 25 2009: (Start)
a( p*10^k ) = p for any prime p.
Proof: if gcd( p, 2*5) = 1, then a( p * 10^k ) = R( R(p) * R(2)^k * R(5)^k ) = R( R(p) * 10^k ) = R(R(p)) = p;
if gcd(p, 2*5) = 2, then p=2 and a( p * 10^k ) = R( R(2)^(k+1) * R(5)^k ) = R( 2 * 10^k ) = 2 = p and mutatis mutandis for gcd(p, 2*5) = 5. (End)

Simpler definition from R. J. Mathar, Jun 16 2009

Edited by N. J. A. Sloane, Jun 23 2009

A102112 Iccanobirt numbers (2 of 15): a(n) = a(n-1) + R(a(n-2)) + a(n-3), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 62, 117, 167, 940, 1818, 2034, 11155, 17275, 74420, 142846, 162568, 885229, 1893336, 2978492, 10197702, 15039830, 38797423, 52888176, 100407789, 206394037, 1246986214, 2077887605, 6411178063, 12726051979

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A102111-A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=a[n-1]+R[a[n-2]]+a[n-3];Table[a[n], {n, 0, 40}]
nxt[{a_,b_,c_}]:={b,c,a+FromDigits[Reverse[IntegerDigits[b]]]+c}; Transpose[ NestList[nxt,{0,0,1},40]][[1]] (* Harvey P. Dale, Jul 29 2013 *)

A004086(a(n)) = A102120(n).

A102124 Iccanobirt numbers (14 of 15): a(n) = R(R(a(n-1)) + R(a(n-2)) + a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 31, 42, 44, 99, 581, 823, 216, 1251, 6592, 3964, 98, 47311, 72451, 99862, 73698, 789881, 684873, 171146, 8359081, 2855313, 6626115, 92901661, 80528542, 25591874, 127303561, 518156392, 14745484, 711014964, 521206301

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A102111-A102125.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, binomial(n, 2),
       R(R(a(n-1)) + R(a(n-2)) + a(n-3)) )
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 18 2014

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[R[a[n-1]]+R[a[n-2]]+a[n-3]];Table[a[n], {n, 0, 40}]

a(n) = A004086(A102116(n)).

A062917 Nonpalindromic numbers k such that k is not divisible by 10 and k*R(k) is a square, where R(k) is the reversal of k (A004086).

Original entry on oeis.org

144, 169, 288, 441, 528, 768, 825, 867, 882, 961, 1089, 1584, 2178, 4851, 8712, 9801, 10404, 10609, 10989, 12544, 12769, 13104, 14544, 14884, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 40401, 44521, 44541, 48139, 48841, 48951

Offset: 1

Amarnath Murthy, Jul 02 2001

The sequence has infinitely many terms, some of which can be derived from 1089 and 2178.

			2178 is a term as 2178*8712 = 4356^2.
10891089 is a term as 10891089*98019801 = 32673267^2.

Giovanni Resta, Table of n, a(n) for n = 1..2338 (terms < 10^12, n = 1..150 from Harvey P. Dale, n = 151..1000 from Chai Wah Wu)
Eric Weisstein's World of Mathematics, Reversal

Cf. A004086.

Subsequence of A029742.

okQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn];idn!= ridn && !Divisible[n,10]&&IntegerQ[Sqrt[n FromDigits[ridn]]]]; Select[Range[ 50000], okQ] (* Harvey P. Dale, Dec 08 2012 *)

More terms from Reiner Martin, Jul 10 2001

A102113 Iccanobirt numbers (3 of 15): a(n) = a(n-1) + R(a(n-2)) + R(a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 62, 135, 203, 760, 1593, 1962, 5980, 12622, 16208, 39724, 142606, 265660, 914694, 1587497, 2150478, 10594748, 27283111, 120773124, 216660897, 649176190, 1868619823, 2758358381, 6139199008, 11266906261

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A102111-A102125.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, binomial(n, 2),
       a(n-1) + R(a(n-2)) + R(a(n-3)) )
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 18 2014

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=a[n-1]+R[a[n-2]]+R[a[n-3]];Table[a[n], {n, 0, 40}]
nxt[{a1_,a2_,a3_}]:={a2,a3,a3+FromDigits[Reverse[IntegerDigits[ a1]]]+ FromDigits[ Reverse[ IntegerDigits[a2]]]}; Transpose[NestList[nxt,{0,0,1},40]][[1]] (* Harvey P. Dale, Oct 17 2012 *)
nxt[{a_,b_,c_}]:={b,c,c+IntegerReverse[b]+IntegerReverse[a]}; NestList[ nxt,{0,0,1},40][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 10 2016 *)

A004086(a(n)) = A102121(n).

A102115 Iccanobirt numbers (5 of 15): a(n) = R(a(n-1)) + a(n-2) + R(a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 42, 44, 117, 779, 1138, 9801, 3204, 22135, 57415, 77633, 144214, 541549, 1123036, 7257201, 3095708, 21636315, 55486847, 104580673, 482935860, 247988412, 1073911003, 3317721397, 9220077878, 15106615327, 89503015162

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A102111-A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[a[n-1]]+a[n-2]+R[a[n-3]];Table[a[n], {n, 0, 40}]
nxt[{a_,b_,c_}]:={b,c,Total[FromDigits/@Reverse/@IntegerDigits[ {a,c}]]+b}; Transpose[NestList[nxt,{0,0,1},35]][[1]] (* Harvey P. Dale, Dec 19 2011 *)

A004086(a(n)) = A102123(n).

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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A102111 Iccanobirt numbers (1 of 15): a(n) = a(n-1) + a(n-2) + R(a(n-3)), where R is the digit reversal function A004086.

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A102125 Iccanobirt numbers (15 of 15): a(n) = R(R(a(n-1)) + R(a(n-2)) + R(a(n-3))), where R is the digit reversal function A004086.

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A061205 a(n) = n times R(n) where R(n) (A004086) is the digit reversal of n.

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A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.

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A102112 Iccanobirt numbers (2 of 15): a(n) = a(n-1) + R(a(n-2)) + a(n-3), where R is the digit reversal function A004086.

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A102124 Iccanobirt numbers (14 of 15): a(n) = R(R(a(n-1)) + R(a(n-2)) + a(n-3)), where R is the digit reversal function A004086.

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