A070837 -id:A070837 - cp's OEIS frontend

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

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 *)

A070838 Smallest prime p such that |p - R(p)| = 9n, where R(n) is digit reversal of n, A004086; or 0 if no such prime exists.

Original entry on oeis.org

23, 13, 41, 37, 61, 17, 29, 19, 0, 1231, 211, 0, 0, 0, 0, 0, 0, 0, 0, 1021, 2903, 103, 0, 0, 0, 0, 0, 0, 0, 1031, 2081, 0, 401, 0, 0, 0, 0, 0, 0, 1151, 4073, 0, 0, 307, 0, 0, 0, 0, 0, 1051, 2281, 0, 0, 0, 227, 0, 0, 0, 0, 1061, 2161, 0, 0, 0, 0, 107, 0, 0, 0, 1181, 2371, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy, May 12 2002

Cf. A070837.

Corrected and extended by Sascha Kurz, Jan 02 2003

A383887 Smallest non-palindromic number that is congruent to its reverse mod n.

Original entry on oeis.org

10, 13, 10, 15, 16, 13, 18, 19, 10, 1011, 100, 15, 1017, 1027, 16, 1025, 1039, 13, 1048, 1021, 18, 103, 1026, 19, 1026, 1017, 14, 1033, 1013, 1011, 1068, 1049, 100, 1039, 1046, 15, 1000, 1055, 1017, 1041, 1066, 1027, 1048, 105, 16, 1077, 1032, 1025, 1014, 1051, 1039, 1017, 1103, 17, 106, 1065

Offset: 1

Erick B. Wong, May 29 2025, at the suggestion of Lanny Wong

Comparable to A070837, but such a number provably exists: if n is coprime to 10 then take 10^k where k is the order of 10 mod n; else take a>b>0 such that n divides 10^a - 10^b, then the number 10^(a+b) + 10^b + 1 works.

The n-th term in this sequence is equal to the first nonzero term of A070837 that occurs at an index divisible by n/gcd(n,9).

			For n=6, a(6)=13 is congruent to 31 mod 6.
For n=10, note that any number of 3 or fewer digits is necessarily a palindrome if the first digit equals the last, and 1011 is the first 4-digit non-palindrome.

Erick B. Wong, Table of n, a(n) for n = 1..10000

Cf. A004086, A056965, A070837.

seq[len_] := Module[{s = Table[0, {len}], c = 0, k = 9, d}, While[c < len, k++; krev = IntegerReverse[k]; If[k != krev, d = Select[Divisors[Abs[k - krev]], # <= len &]; Do[If[s[[d[[i]]]] == 0, s[[d[[i]]]] = k; c++], {i, 1, Length[d]}]]]; s]; seq[60] (* Amiram Eldar, May 31 2025 *)

a(n) = my(k=1, d=digits(k), rd=Vecrev(d)); while(!((d != rd) && Mod(fromdigits(rd), n) == k), k++; d=digits(k); rd=Vecrev(d)); k; \\ Michel Marcus, May 30 2025

from functools import partial
from itertools import count
def accept(mod, k):
    r = int(str(k)[::-1])
    return r != k and (r-k) % mod == 0
def a(n):
    return next(filter(partial(accept, n), count(10)))
print([a(n) for n in range(1,57)])

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A070838 Smallest prime p such that |p - R(p)| = 9n, where R(n) is digit reversal of n, A004086; or 0 if no such prime exists.

Views

Author

Keywords

Crossrefs

Extensions

A383887 Smallest non-palindromic number that is congruent to its reverse mod n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python