A066058 -id:A066058 - cp's OEIS frontend

A066057 'Reverse and Add' carried out in base 2 (cf. A062128); number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 4, 5, 0, -1, 2, 1, 4, -1, 0, -1, 2, 1, 0, 1, 0, 1, -1, 1, -1, 1, 2, 1, -1, 1, 2, 3, 0, -1, -1, 1, -1, 3, 0, 1, 2, 3, 2, 1, 2, 3, 2, -1, -1, 1, 0, 1, 0, 1, -1, 1, 2, 1, 4, 3, 0, 11, -1, 5, -1, -1, 2, 1, 2, 1, 4, -1, 0, -1, 2, 5, -1, -1, 2, 3, 0, -1, -1, 1, -1, 3, 0, 1, 4, 1, 10, 11, -1, -1, 0, -1, 2, -1, 4

Offset: 0

Klaus Brockhaus, Dec 04 2001

The analog of A033665 in base 2.

			10011 (19 in base 10) -> 10011 + 11001 = 101100 -> 101100 + 1101 = 111001 -> 111001 + 100111 = 1100000 -> 1100000 + 11 = 1100011 (palindrome) requires 4 steps, so a(19) = 4.

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

Cf. A033665, A058042, A062128, A062130, A033865, A061561, A066058, A006995, A057148.

function b2reverse(a: integer): integer; var n,i,rev: integer; begin n := bit_length(a); for i := 0 to n-1 do if bit_test(a,i) = 1 then rev := bit_set(rev,n-1-i); end; end; return rev; end; function a066057(mx,stop: integer); var c,k,m,rev: integer; begin for k := 0 to mx do c := 0; m := k; rev := b2reverse(m); while m <> rev and c < stop do inc(c); m := m + rev; rev := b2reverse(m); end; if c < stop then write(c); else write(-1); end; write(" "); end; end; a066057(120,300);

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, i], {n, 0, 111}] (* Robert Price, Oct 14 2019 *)

A066144 In base 2: n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.

Original entry on oeis.org

0, 2, 11, 19, 20, 74, 398, 779, 1062, 2329, 4189, 4280, 11278, 19962, 98318, 135137, 1051360, 1592930, 69226926, 4295054186, 4446008678, 17187271449, 18123849698

Offset: 1

Klaus Brockhaus, Dec 08 2001

The analog of A065198 in base 2. Integers like 22, for which a palindrome is never reached (cf. A066059), are of course disregarded. A066145 gives the corresponding records.

			Starting with 74, 11 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 74, less (at most 5) steps are needed.

Index entries for sequences related to Reverse and Add!

Record setting values in base b: A077406 (b=3), A075686 (b=4), A306599 (b=8), A065198 (b=10), A348571 (Zeckendorf).

Cf. A062130, A066057, A066058, A066059, A066145.

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; Select[Range[0, 100000], (np = #; i = 0;
   While[np != IntegerReverse[np, 2] && i < limit,
    np = np + IntegerReverse[np, 2]; i++];
If[i >= limit, False, If[i > best, best = i; True]]) &] (* Robert Price, Oct 14 2019 *)

Offset corrected and a(19)-a(23) from A.H.M. Smeets, Apr 30 2022

A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 4, 5, 11, 21, 32, 37, 46, 48, 49, 53, 89, 99, 142, 147, 273, 297, 345, 515, 550, 573

Offset: 1

Klaus Brockhaus, Dec 08 2001

The analog of A065199 in base 2. A066144 gives the corresponding starting points.

Terms a(19..22) obtained by assuming that a(n+1) <= a(n) + 300. - A.H.M. Smeets, Apr 30 2022

			Starting with 74, 11 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 74, at most 5 steps are needed.

Index entries for sequences related to Reverse and Add!

Record values in base b: A077407 (b=3), A075687 (b=4), A306600 (b=8), A065199 (b=10), A348572 (Zeckendorf).

Cf. A062130, A066057, A066058, A066059, A066144.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; lst = {};
For[n = 0, n <= 10000, n++,
np = n; i = 0;
While[np != IntegerReverse[np, 2] && i < limit,
  np = np + IntegerReverse[np, 2]; i++];
If[i < limit && i > best, best = i; AppendTo[lst, i]]]; lst (* Robert Price, Oct 14 2019 *)

Offset corrected and a(19)-a(23) by A.H.M. Smeets, Apr 30 2022

A077441 In base 4, smallest number that requires n Reverse and Add! steps to reach a palindrome.

Original entry on oeis.org

0, 4, 7, 26, 28, 127, 306, 348, 398, 301, 308, 203, 311, 783, 294, 350, 199, 296, 4268, 16595, 5326, 4253, 17399, 8235, 6189, 4270, 3107, 1270, 1532, 511, 67816, 65975, 24670, 12395, 4282, 3119, 28799, 16861, 18164, 66268, 45087, 71164, 309234

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base-4 analog of A066058 (base 2) and A023109 (base 10).

			7 is the smallest number which requires two steps to reach a base 4 palindrome (cf. A075685), so a(2) = 5; 7 (decimal) = 13 -> 13 + 31 = 110 -> 110 + 011 = 121 (palindrome) = 25 (decimal).

Cf. A075685, A066058, A023109.

{m=46; v=[]; for(j=1,m+1,v=concat(v,-1)); mc=m+1; n=0; while(mc>0,a=-1; c=0; k=n; while(c0,d=divrem(q,4); q=d[1]; rev=4*rev+d[2]); if(k==rev,a=c; c=m+1,c++; k=k+rev)); if(0<=a&&a<=m,if(v[a+1]<0,v[a+1]=n; mc--; print1([a,n]))); n++); print(); for(j=1,m+1,print1(v[j],","))}

from gmpy2 import digits
def A077441(n):
    if n > 0:
        k = 0
        while True:
            m = k
            for i in range(n):
                s = digits(m,4)
                if s == s[::-1]:
                    break
                m += int(s[::-1],4)
            else:
                s = digits(m,4)
                if s == s[::-1]:
                    return k
            k += 1
    else:
        return 0 # Chai Wah Wu, Jan 17 2015

A077403 In base 3: smallest number that requires n Reverse and Add! steps to reach a palindrome.

Original entry on oeis.org

0, 3, 5, 15, 17, 263, 170, 509, 491, 322, 266, 222, 161, 494, 260, 106, 95, 78, 53, 2425, 1466, 9717, 59583, 38878, 38798, 33515, 39440, 32857, 37340, 238849, 177470, 60019, 59655, 178540, 124895, 59753, 179751, 1595576, 715615, 354605, 179575

Offset: 0

Klaus Brockhaus, Nov 05 2002

Base 3 analog of A066058 (base 2), A077441 (base 4) and A023109 (base 10).

			5 is the smallest number which requires two steps to reach a base 3 palindrome (cf. A066057), so a(2) = 5; 5 (decimal) = 12 -> 12 + 21 = 110 -> 110 + 011 = 121 (palindrome) = 16 (decimal).

Index entries for sequences related to Reverse and Add!

Cf. A066057, A066058, A077441, A023109.

var ar: array; end; m := 45; ar := alloc(array, m+1, -1); mc := m+1; n := 0; while mc > 0 do v := -1; c := 0; k := n; while c < m+1 do d := k; rev := 0; while d > 0 do rev := 3*rev+(d mod 3); d := d div 3; end; if k = rev then v := c; c := m+1; else inc(c); k := k+rev; end; end; if 0 <= v and v <= m then if ar[v] < 0 then ar[v] := n; dec(mc); write((v,n)); end; end; inc(n); end; writeln(); for j := 0 to m do write(ar[j],","); end;

cp's OEIS Frontend

A066057 'Reverse and Add' carried out in base 2 (cf. A062128); number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

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Mathematica

A066144 In base 2: n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.

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Mathematica

Extensions

A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

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Mathematica

Extensions

A077441 In base 4, smallest number that requires n Reverse and Add! steps to reach a palindrome.

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PARI

Python

A077403 In base 3: smallest number that requires n Reverse and Add! steps to reach a palindrome.

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ARIBAS