A056965 -id:A056965 - cp's OEIS frontend

A056964 a(n) = n + reversal of digits of n.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88, 99, 110

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits then a(n) is a multiple of 11.

Also called the Reverse and Add!, or RADD operation. Iteration of this function leads to the definition of Lychrel and related numbers, cf. A023108, A063048, A088753, A006960, and many others. - M. F. Hasler, Apr 13 2019

			a(17) = 17 + 71 = 88.

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Reverse-Then-Add Sequence
Index entries for sequences related to Reverse and Add!

Cf. A004086, A056965, A067030.
Differs from A052008 when n=101 and a(101)=202 while A052008(101)=121
Cf. A036839.

a056964 n = n + a004086 n  -- Reinhard Zumkeller, Oct 14 2011

Table[n+FromDigits[Reverse[IntegerDigits[n]]],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
#+IntegerReverse[#]&/@Range[0,100] (* Harvey P. Dale, Jul 31 2025 *)

A056964(n)=fromdigits(Vecrev(digits(n)))+n \\ Charles R Greathouse IV, Oct 28 2014

def A056964(n): return n+int(str(n)[::-1]) # Indranil Ghosh, Jan 29 2017

a(n) = n + A004086(n) = 2*n - A056965(n).

n < a(n) < 11n for n > 0. - Charles R Greathouse IV, Nov 17 2022

A072137 Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 6, 4, 5, 3, 3, 5, 4, 6, 2, 1, 2, 1, 2, 6, 4

Offset: 0

Klaus Brockhaus, Jun 24 2002

'Reverse and Subtract' (cf. A070837, A070838) is defined by x -> |x - reverse(x)|, where reverse(x) is the digit reversal of x.
For every n the trajectory eventually becomes periodic, since 'Reverse and Subtract' does not increase the number of digits and so the set of available terms is finite. For small n the period length is 1, the periodic part consists of 0's, the last term of the preperiodic part is a palindrome.
The first n with period length 2 and a nontrivial periodic part is 1012 (cf. A072140).
This sequence is a weak analog of A033665, which uses 'Reverse and Add'.

			a(15) = 4 since 15 -> |15- 51| = 36 -> |36 - 63| = 27 -> |27 - 72| = 45 -> |45 - 54| = 9.

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A033665, A070837, A070838, A072138, A072139, A072140, A072141, A072146, A072147.

import Data.List(inits, find); import Data.Maybe(fromJust)
a072137 :: Int -> Int
a072137 = length . fst . spanCycle (abs . a056965) where
   spanCycle :: Eq a => (a -> a) -> a -> ([a],[a])
   spanCycle f x = fromJust $ find (not . null . snd) $
                              zipWith (span . (/=)) xs $ inits xs
                   where xs = iterate f x
-- Reinhard Zumkeller, Oct 24 2010

a[n_] := (k = 0; FixedPoint[ (k++; Abs[# - FromDigits[ Reverse[ IntegerDigits[#] ] ] ]) &, n]; k - 1); Table[ a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 01 2011 *)

A072140 The period length of the 'Reverse and Subtract' trajectory of n is greater than 1.

Original entry on oeis.org

1012, 1023, 1034, 1045, 1067, 1078, 1089, 1100, 1122, 1133, 1144, 1155, 1177, 1188, 1199, 1210, 1232, 1243, 1254, 1265, 1287, 1298, 1320, 1342, 1353, 1364, 1375, 1397, 1408, 1430, 1452, 1463, 1474, 1485, 1507, 1518, 1540, 1562, 1573, 1584, 1595, 1606

Offset: 1

Klaus Brockhaus, Jun 24 2002

'Reverse and Subtract' (cf. A072137) is defined by x -> |x - reverse(x)|. There is no number k > 0 such that |k - reverse(k)| = k, so 0 is the only period with length 1. Consequently this sequence consists of the numbers n such that repeated application of 'Reverse and Subtract' does not lead to a palindrome. It is an analog of A023108, which uses 'Reverse and Add'. - A072141, A072142, A072143 give the numbers which generate periods of length 2, 14, 22 respectively.

			1012 -> |1012 - 2101| = 1089 -> |1089 - 9801| = 8712 -> |8712 - 2178| = 6534 -> |6534 - 4356| = 2178 -> |2178 - 8712| = 6534; the period of the trajectory is 6534, 2178 and a palindrome is never reached.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A023108, A072137, A072141, A072142, A072143.

import Data.List (find, findIndices, inits)
import Data.Maybe (fromJust)
spanCycle :: Eq a => (a -> a) -> a -> ([a],[a])
spanCycle f x = fromJust $ find (not . null . snd) $
                           zipWith (span . (/=)) xs $ inits xs
                where xs = iterate f x
a072140_list = findIndices (> 1) $
               map (length . snd . spanCycle (abs . a056965)) [0..]
-- eop.
-- Reinhard Zumkeller, Oct 24 2010

A068396 n-th prime minus its reversal.

Original entry on oeis.org

0, 0, 0, 0, 0, -18, -54, -72, -9, -63, 18, -36, 27, 9, -27, 18, -36, 45, -9, 54, 36, -18, 45, -9, 18, 0, -198, -594, -792, -198, -594, 0, -594, -792, -792, 0, -594, -198, -594, -198, -792, 0, 0, -198, -594, -792, 99, -99, -495

Offset: 1

Reinhard Zumkeller, Mar 08 2002

a(n) = 0 for n in A075807. - Michel Marcus, Sep 27 2017

All terms are divisible by 9. - Zak Seidov, Jun 05 2021

			a(10) = 29 - 92 = -63;
a(20) = 71 - 17 = 54.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A004086, A056965, A061227, A075807.

a068396 n = p - a004086 p  where p = a000040 n
-- Reinhard Zumkeller, Feb 04 2014

#-IntegerReverse[#]& /@ Prime[Range[50]] (* Harvey P. Dale, Dec 20 2012 *)

a(n) = prime(n) - fromdigits(Vecrev(digits(prime(n)))); \\ Michel Marcus, Sep 27 2017

from sympy import prime
def a(n): pn = prime(n); return pn - int(str(pn)[::-1])
print([a(n) for n in range(1, 50)]) # Michael S. Branicky, Jun 05 2021

a(n) = A000040(n) - A004087(n).

a(n) = A056965(A000040(n)). - Michel Marcus, Sep 27 2017

A070837 Smallest number k such that abs(k - R(k)) = 9n, where R(k) is digit reversal of k (A004086); or 0 if no such k exists.

Original entry on oeis.org

10, 13, 14, 15, 16, 17, 18, 19, 90, 1011, 100, 0, 0, 0, 0, 0, 0, 0, 0, 1021, 1090, 103, 0, 0, 0, 0, 0, 0, 0, 1031, 1080, 0, 104, 0, 0, 0, 0, 0, 0, 1041, 1070, 0, 0, 105, 0, 0, 0, 0, 0, 1051, 1060, 0, 0, 0, 106, 0, 0, 0, 0, 1061, 1050, 0, 0, 0, 0, 107, 0, 0, 0, 1071, 1040, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy, May 12 2002

If 9n < 1000, k has at most 5 digits, and abs(k - R(k)) = 9n, then 10 must divide n. - Sascha Kurz, Jan 02 2003

Cf. A004086, A056965, A070838.

a = Table[0, {50}]; Do[d = Abs[n - FromDigits[ Reverse[ IntegerDigits[n]]]] / 9; If[d < 51 && a[[d]] == 0, a[[d]] = n], {n, 1, 10^7}]; a

def back_difference(n):
    r = int(str(n)[::-1])
    return abs(r-n)
def a070837(n):
    i = 0
    while True:
        if back_difference(i)==9*n:
            return i
        i+=1
# Christian Perfect, Jul 23 2015

More terms from Sascha Kurz, Jan 02 2003

a(21) corrected by Christian Perfect, Jul 23 2015

A256756 a(n) = bitwise XOR of n and the reverse of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 25, 18, 39, 60, 45, 86, 67, 72, 22, 25, 0, 55, 50, 45, 36, 83, 78, 65, 29, 18, 55, 0, 9, 22, 27, 108, 117, 122, 44, 39, 50, 9, 0, 27, 110, 101, 100, 111, 55, 60, 45, 22, 27, 0, 121, 114, 111, 100, 58, 45, 36, 27, 110, 121

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A003987, A004086, A055483, A056964, A056965, A061205, A068634, A175919, A256754, A256755.

a:= n-> Bits[Xor](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitXor[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,65}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitxor(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A003987(n, A004086(n)).

A256754 a(n) = bitwise AND of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 4, 13, 8, 3, 16, 1, 16, 19, 0, 4, 22, 0, 8, 16, 26, 8, 16, 28, 2, 13, 0, 33, 34, 33, 36, 1, 2, 5, 0, 8, 8, 34, 44, 36, 0, 10, 16, 16, 0, 3, 16, 33, 36, 55, 0, 9, 16, 27, 4, 16, 26, 36, 0, 0, 66, 64, 68, 64, 6, 1, 8, 1, 10

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A004086, A004198, A055483, A056964, A056965, A061205, A068634, A256755, A256756.

a:= n-> Bits[And](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitAnd[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,74}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitand(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A004198(n,A004086(n)).

A256755 a(n) = bitwise OR of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 29, 31, 47, 63, 61, 87, 83, 91, 22, 29, 22, 55, 58, 61, 62, 91, 94, 93, 31, 31, 55, 33, 43, 55, 63, 109, 119, 127, 44, 47, 58, 43, 44, 63, 110, 111, 116, 127, 55, 63, 61, 55, 63, 55, 121, 123, 127, 127, 62, 61, 62, 63, 110

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A003986, A004086, A055483, A056964, A056965, A061205, A068634, A256754, A256756.

a:= n-> Bits[Or](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitOr[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,64}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitor(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A003986(n,A004086(n)).

A347688 Let c (resp. C) be the smallest (resp. largest) number that can be obtained by cyclically permuting the digits of n; a(n) = C - c.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 9, 18, 27, 36, 45, 54, 63, 72, 18, 9, 0, 9, 18, 27, 36, 45, 54, 63, 27, 18, 9, 0, 9, 18, 27, 36, 45, 54, 36, 27, 18, 9, 0, 9, 18, 27, 36, 45, 45, 36, 27, 18, 9, 0, 9, 18, 27, 36, 54, 45, 36, 27, 18, 9, 0, 9, 18, 27, 63, 54, 45, 36, 27, 18, 9, 0, 9, 18, 72, 63, 54, 45, 36, 27, 18, 9, 0, 9, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 99, 99, 189, 279, 369, 459, 549, 639, 729, 819, 99, 0, 99

Offset: 0

N. J. A. Sloane, Sep 22 2021, following a suggestion from Joseph Rozhenko

Agrees with A151949 for n <= 101.
All terms are multiples of 9 (cf. A347689).
Repeatedly applying the operation of subtracting the smallest cyclic permutation from the largest produces interesting sequences. It appears that n-digit numbers converge to a small number m << n of loops. Any 3-digit number converges to one of two loops of length 3: (189, 729, 675) and (378, 459, 486); any 4-digit number to either of (189, 729, 675) or a loop of length 25; any 5-digit number to one of three loops of length 3: (38007, 79335, 59778), (48780, 82926, 65853), or (29889, 69003, 89667), and so on. For 6-digit numbers there are 10 loops (of lengths 1, 2, 3, 10, 12 or 60). - Joseph Rozhenko, Oct 05 2021
For any cyclic number N, there must exist a cyclic permutation P of N for which a(P) = P. - Joseph Rozhenko, Oct 07 2021
An interesting plot of this sequence comes up when the axes are not n vs. a(n), but rather C vs. c. In other words, each number a(n) is represented on the X axis by C (largest number that can be obtained by cyclically permuting the digits of n) and on the Y axis as c (smallest number that can be obtained by cyclically permuting the digits of n). - Joseph Rozhenko, Jan 26 2022

			If n = 102, c = 21, C = 210, a(102) = 210 - 21 = 189.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A010785, A056965, A151949, A163380, A163381, A347689.

A347688 := proc(n)
        local dgs,C,c,ndgs,r ;
        dgs := convert(n,base,10) ;
        ndgs := nops(dgs) ;
        C := digcatL(dgs) ;
        c := C ;
        for r from 0 to ndgs-1 do
                C := max(C,digcatL(dgs)) ;
                c := min(c,digcatL(dgs)) ;
                dgs := ListTools[Rotate](dgs,1) ;
        end do:
        C-c ;
end proc: # R. J. Mathar, Sep 27 2021

{0}~Join~Table[First@Differences@MinMax[FromDigits/@(RotateLeft[IntegerDigits@n,#]&/@Range@IntegerLength@n)],{n,112}] (* Giorgos Kalogeropoulos, Sep 22 2021 *)

a(n, base=10) = { my (c=n, C=n, d=digits(n, base)); for (k=1, #d, my (r=fromdigits(concat(d[k+1..#d], d[1..k]), base)); c=min(c, r); C=max(C, r)); C-c } \\ Rémy Sigrist, Sep 22 2021

def a(n):
    s = str(n)
    cyclic_perms = [int("".join(s[i:] + s[:i])) for i in range(len(s))]
    c, C = min(cyclic_perms), max(cyclic_perms)
    return C - c
print([a(n) for n in range(113)]) # Michael S. Branicky, Sep 26 2021

a(n) = 0 iff n belongs to A010785. - Rémy Sigrist, Sep 22 2021

More than the usual number of terms are shown in order to distinguish this from similar sequences.

A055947 n - reversal of base 3 digits of n (written in base 10).

Original entry on oeis.org

0, 0, 0, 2, 0, -2, 4, 2, 0, 8, 0, -8, 8, 0, -8, 8, 0, -8, 16, 8, 0, 16, 8, 0, 16, 8, 0, 26, 0, -26, 20, -6, -32, 14, -12, -38, 32, 6, -20, 26, 0, -26, 20, -6, -32, 38, 12, -14, 32, 6, -20, 26, 0, -26, 52, 26, 0, 46, 20, -6, 40, 14, -12, 58, 32, 6, 52, 26, 0, 46, 20, -6, 64, 38, 12, 58, 32, 6, 52, 26, 0, 80, 0, -80, 56, -24, -104, 32, -48

Offset: 0

Henry Bottomley, Jul 18 2000

a(n) is even.

			For n = 5, the reversal of base 3 digits of n (written in base 10) is 7. So, a(5) = 5 - 7 = -2. - _Indranil Ghosh_, Feb 01 2017

Indranil Ghosh, Table of n, a(n) for n = 0..20000

Cf. A030102, A056965.

Table[n-FromDigits[Reverse[IntegerDigits[n,3]],3],{n,0,90}] (* Harvey P. Dale, May 14 2022 *)

a(n) = n - fromdigits(Vecrev(digits(n, 3)), 3); \\ Michel Marcus, Aug 09 2019

a(n) = n - A030102(n).

cp's OEIS Frontend

A056964 a(n) = n + reversal of digits of n.

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A072137 Length of the preperiodic part of the 'Reverse and Subtract' trajectory of n.

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A072140 The period length of the 'Reverse and Subtract' trajectory of n is greater than 1.

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A068396 n-th prime minus its reversal.

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A070837 Smallest number k such that abs(k - R(k)) = 9n, where R(k) is digit reversal of k (A004086); or 0 if no such k exists.

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A256756 a(n) = bitwise XOR of n and the reverse of n.

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A256754 a(n) = bitwise AND of n and the reverse of n.

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