A033865 -id:A033865 - cp's OEIS frontend

A067032 Number of k's such that A067030(n) = k + reverse(k).

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

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 3 since A067030(12) = 33 and for k = 12, 21, 30 we have 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067030, A067031, A067033, A067034.

function a067032(a,b: integer); var n,k,c,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; c := 0; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then inc(c); end; inc(k); end; if c > 0 then write(c,","); end; end; end; a067032(0,1000);

A063049 Integers n > 196 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 196.

Original entry on oeis.org

295, 394, 493, 592, 689, 691, 788, 790, 887, 986, 1495, 1585, 1675, 1765, 1855, 1945, 2494, 2584, 2674, 2764, 2854, 2944, 3493, 3583, 3673, 3763, 3853, 3943, 4079, 4169, 4259, 4349, 4439, 4492, 4529, 4582, 4619, 4672, 4709, 4762, 4799, 4852, 4889, 4942

Offset: 1

Klaus Brockhaus, Jul 07 2001

Subsequence of A023108.

			The trajectory of 394 reaches 887 in one step and 887 is a term in the trajectory of 196, so 394 belongs to the present sequence. The corresponding term in A063050, giving the number of steps, accordingly is 1.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), page PC30-9. Gives initial terms of this sequence.

Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063050.

Block[{nn = 10^2, s}, s = NestList[# + IntegerReverse@ # &, 196, nn]; Rest@ Select[Range@ 5000, Length@NestWhileList[# + IntegerReverse@ # &, #, FreeQ[s, #] &, 1, nn] <= nn &]] (* Michael De Vlieger, Jan 21 2018 *)

Offset corrected by Sean A. Irvine, Apr 17 2023

A067034 Largest k such that A067030(n) = k + reverse(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 7, 8, 9, 20, 30, 40, 50, 60, 70, 80, 90, 100, 91, 110, 93, 120, 94, 95, 130, 96, 97, 140, 98, 99, 150, 200, 160, 210, 170, 220, 180, 230, 190, 240, 250, 300, 260, 310, 270, 320, 280, 330, 290, 340, 350, 400, 360, 410, 370, 420, 380, 430

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 30 since A067030(12) = 33 and 30 is the largest k such that 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067031, A067031, A067032, A067033.

function a067034(a,b: integer); var n,k,m,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; m := -1; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then m := k; end; inc(k); end; if m >= 0 then write(m,","); end; end; end; a067034(0,500);

A061563 Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 33, 55, 77, 99, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 44, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 66, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 88, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 121, 121, 363, 484, 1111, 66, 77, 88, 99, 121

Offset: 0

N. J. A. Sloane, May 18 2001

It is believed that n = 196 is the smallest integer which never reaches a palindrome.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

Cf. A033865. A016016 (number of steps), A023109, A006950, A023108.

var st: stack; test: boolean; end; for k := 0 to 60 do n := k; test := true; while test do n := n + int_reverse(n); test := n <> int_reverse(n); end; stack_push(st,n); end; stack2array(st);

tol = 1000; r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, n = -1]; n, {n, 0, 64}] (* Jayanta Basu, Jul 11 2013 *)
Table[Module[{k=n+IntegerReverse[n]},While[k!=IntegerReverse[k],k=k+IntegerReverse[k]];k],{n,0,70}] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jul 19 2016 *)

Corrected and extended by Klaus Brockhaus, May 20 2001

More terms from Ray Chandler, Jul 25 2003

A067033 Smallest k such that A067030(n) = k + reverse(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 100, 19, 29, 39, 120, 49, 59, 130, 69, 79, 140, 89, 99, 150, 101, 160, 111, 170, 121, 180, 131, 190, 141, 151, 102, 161, 112, 171, 122, 181, 132, 191, 142, 152, 103, 162, 113, 172, 123, 182, 133, 192

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 12 since A067030(12) = 33 and 12 is the smallest k such that 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067030, A067031, A067032, A067034.

function a067033(a,b: integer); var n,k,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then write(k,",") k := n + 1; else inc(k); end; end; end; end; a067033(0,500);

A062130 A062128 written in base 10.

Original entry on oeis.org

0, 1, 3, 3, 5, 5, 9, 7, 9, 9, 15, 27, 15, 27, 21, 15, 17, 17, 27, 99, 99, 21, -1, 63, 27, 99, -1, 27, -1, 63, 45, 31, 33, 33, 51, -1, 45, -1, 63, 99, 45, -1, 63, 99, 99, 45, -1, -1, 51, -1, 255, 51, 63, 99, 255, 153, 63, 99, 255, 153, -1, -1, 93, 63, 65, 65, 99, -1, 85, 255, 119, 387, 255, 73, 13299, -1, 387, -1, -1, 219, 85

Offset: 0

Klaus Brockhaus, Jun 06 2001

			23 -> 23 + 29 = 52 -> 52 + 11 = 63, so a(23) = 63.

Index entries for sequences related to Reverse and Add!
Klaus Brockhaus, On the'Reverse and Add!' algorithm in base 2

Cf. A033865, A062128, A062131, A061561.

stop := 500; for k := 0 to 80 do c := 0; m := k; rev := bit_reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := bit_reverse(m); end; if c < stop then write(m); else write(-1); end; write(" "); end;.

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Table[np = n; i = 0;
 While[np != IntegerReverse[np, 2] && i < limit,
  np = np + IntegerReverse[np, 2]; i++];
If[i >= limit, -1, np], {n, 0, 80}] (* Robert Price, Oct 14 2019 *)

A066058 In base 2: smallest integer which requires n 'Reverse and Add' steps to reach a palindrome.

Original entry on oeis.org

0, 2, 11, 44, 19, 20, 275, 326, 259, 202, 103, 74, 1027, 1070, 1049, 1072, 1547, 1310, 1117, 794, 569, 398, 3083, 2154, 1177, 1064, 4697, 4264, 4443, 2678, 2169, 1422, 779, 3226, 1551, 1114, 1815, 1062, 4197, 3106, 8697, 7238, 16633, 12302, 6683

Offset: 0

Klaus Brockhaus, Dec 04 2001

The analog of A023109 in base 2.

			11 is the smallest integer which requires two steps to reach a base 2 palindrome (cf. A066057), so a(2) = 11; written in base 10: 11 -> 11 + 13 = 24 -> 24 + 3 = 27; written in base 2: 1011 -> 1011 + 1101 = 11000 -> 11000 + 11 = 11011.

Cf. A066057, A023109, A062128, A062130, A033865, A006995, A057148.

(* For function b2reverse see A066057. *) function a066058(mx: integer); var k,m,n,rev,steps: integer; begin for k := 0 to mx do n := 0; steps := 0; m := n; rev := b2reverse(m); while not(steps = k and m = rev) do inc(n); m := n; rev := b2reverse(m); steps := 0; while steps < k and m <> rev do m := m + rev; rev := b2reverse(m); inc(steps); end; end; write(n,","); end; end; a066058(45);

Table[ SelectFirst[Range[0, 20000], (np = #; i = 0;
    While[ np != IntegerReverse[np, 2] && i <= n,
     np = np + IntegerReverse[np, 2]; i++];
i == n ) &] , {n, 0, 44}] (* Robert Price, Oct 16 2019 *)

def A066058(n):
    if n > 0:
        k = 0
        while True:
            m = k
            for i in range(n):
                s1 = format(m,'b')
                s2 = s1[::-1]
                if s1 == s2:
                    break
                m += int(s2,2)
            else:
                s1 = format(m,'b')
                if s1 == s1[::-1]:
                    return k
            k += 1
    else:
        return 0 # Chai Wah Wu, Jan 06 2015

A062128 In base 2: start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 1, 11, 11, 101, 101, 1001, 111, 1001, 1001, 1111, 11011, 1111, 11011, 10101, 1111, 10001, 10001, 11011, 1100011, 1100011, 10101, -1, 111111, 11011, 1100011, -1, 11011, -1, 111111, 101101, 11111, 100001, 100001, 110011, -1, 101101, -1, 111111, 1100011, 101101, -1, 111111, 1100011, 1100011

Offset: 0

Klaus Brockhaus, Jun 06 2001

The analog of A033865 in base 2.

			23: 10111 -> 10111 + 11101 = 110100 -> 110100 + 1011 = 111111, so a(23) = 111111.

Klaus Brockhaus, On the'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A033865, A062129, A062130, A058042.

stop := 500; for k := 0 to 60 do c := 0; m := k; rev := bit_reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := bit_reverse(m); end; if c < stop then bit_write(m); else write(-1); end; write(" "); end;

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
BaseForm[Table[np = n; i = 0;
  While[np != IntegerReverse[np, 2] && i < limit,
   np = np + IntegerReverse[np, 2]; i++];
If[i >= limit, -1, np], {n, 0, 44}], 2] (* Robert Price, Oct 14 2019 *)

A063433 'Reverse and Add!' trajectory of 10577.

Original entry on oeis.org

10577, 88078, 175166, 836737, 1574375, 7309126, 13528163, 49710694, 99312488, 187733887, 976071668, 1842242347, 9274664828, 17559329557, 93151725128, 175304440267, 937348843838, 1775697687577, 9533565653348, 17967131306707

Offset: 0

Klaus Brockhaus, Jul 20 2001

			a(1) = 10577 + 77501 = 88078.

Harry J. Smith, Table of n, a(n) for n=0,...,200
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063434.

Cf. A056964, A004086.

m := 10577; stop := 20; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063433 n = a063433_list !! n
a063433_list = iterate a056964 10577 -- Reinhard Zumkeller, Sep 22 2011

NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,10577, 20]  (* Harvey P. Dale, Apr 03 2011 *)

Rev(x)= { local(d,r); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=10577); write("b063433.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 21 2009

A063051 'Reverse and Add!' trajectory of 879.

Original entry on oeis.org

879, 1857, 9438, 17787, 96558, 182127, 903408, 1707717, 8884788, 17759676, 85455447, 159910905, 668930856, 1326970722, 3597766953, 7194444906, 13288889823, 46187778054, 91275556218, 172541113437, 906852258708

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 879 + 978 = 1857.

Harry J. Smith and Michael Lee, Table of n, a(n) for n = 0..2366 (Harry J. Smith provided terms 0 through 200)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063052, A056964, A004086.

m := 879; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a033651 n = a033651_list !! n
a033651_list = iterate a056964 9 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 879, 40] (* Vincenzo Librandi, Sep 23 2013 *)

Rev(x)= { local(d); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=879); write("b063051.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 16 2009

cp's OEIS Frontend

A067032 Number of k's such that A067030(n) = k + reverse(k).

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A063049 Integers n > 196 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 196.

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A067034 Largest k such that A067030(n) = k + reverse(k).

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A061563 Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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A067033 Smallest k such that A067030(n) = k + reverse(k).

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A062130 A062128 written in base 10.

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A066058 In base 2: smallest integer which requires n 'Reverse and Add' steps to reach a palindrome.

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A062128 In base 2: start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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