| Displaying 1-9 of 9 results found.
page
1
9999, 99999, 990099, 999999, 9901099, 9905099, 9993999, 9996999, 9997999, 9998999, 9999999, 99999999, 990959099, 990969099, 999010999, 999020999, 999030999, 999040999, 999070999, 999929999, 999939999, 999969999, 999989999
COMMENTS
Palindromes in A088753; palindromes for which the Reverse and Add! process does not lead to another palindrome. The numbers were extracted from W. VanLandingham's list of Lychrel numbers.
Number of palindromes < 10^n in A088753.
+20
3
0, 0, 0, 1, 2, 4, 11, 12, 24, 26, 55
COMMENTS
Number of terms < 10^n in A089521, determined with the aid of W. VanLandingham's list of Lychrel numbers.
Palindromes in base 10.
(Formerly M0484 N0178)
+10
800
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515
COMMENTS
It seems that if n*reversal(n) is in the sequence then n = 3 or all digits of n are less than 3. - Farideh Firoozbakht, Nov 02 2014 The position of a palindrome within the sequence can be determined almost without calculation: If the palindrome has an even number of digits, prepend a 1 to the front half of the palindrome's digits. If the number of digits is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit. Examples: 98766789 = a(19876), 515 = a(61), 8206028 = a(9206), 9230329 = a(10230). - Hugo Pfoertner, Aug 14 2015 The order has been reduced from 49 to 3; see the Cilleruelo-Luca and Cilleruelo-Luca-Baxter links. - Jonathan Sondow, Nov 27 2017 The number of palindromes with d digits is 10 if d = 1, and otherwise it is 9 * 10^(floor((d - 1)/2)). - N. J. A. Sloane, Dec 06 2015 Sequence A033665 tells how many iterations of the Reverse-then-add function A056964 are needed to reach a palindrome; numbers for which this will never happen are Lychrel numbers ( A088753) or rather Kin numbers ( A023108). - M. F. Hasler, Apr 13 2019
REFERENCES
Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
read transforms; t0:=[]; for n from 0 to 2000 do if digrev(n) = n then t0:=[op(t0), n]; fi; od: t0;
# Alternatively, to get all palindromes with <= N digits in the list "Res":
N:=5;
Res:= $0..9:
for d from 2 to N do
if d::even then
m:= d/2;
Res:= Res, seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1);
else
m:= (d-1)/2;
Res:= Res, seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1);
fi
# A variant: Gets all base-10 palindromes with exactly d digits, in the list "Res"
d:=4:
if d=1 then Res:= [$0..9]:
elif d::even then
m:= d/2:
Res:= [seq(n*10^m + digrev(n), n=10^(m-1)..10^m-1)]:
else
m:= (d-1)/2:
Res:= [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
fi:
isA002113 := proc(n)
simplify(digrev(n) = n) ;
MATHEMATICA
palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; (* then to generate any base-b sequence for 1 < b < 37, replace the 10 in the following instruction with b: *) Select[Range[0, 1000], palQ[#, 10] &]
base10Pals = {0}; r = 2; Do[Do[AppendTo[base10Pals, n * 10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}]; Do[AppendTo[base10Pals, n * 10^IntegerLength[n] + FromDigits@Reverse@IntegerDigits[n]], {n, 10^(e - 1), 10^e - 1}], {e, r}]; base10Pals (* Arkadiusz Wesolowski, May 04 2012 *) nthPalindromeBase[n_, b_] := Block[{q = n + 1 - b^Floor[Log[b, n + 1 - b^Floor[Log[b, n/b]]]], c = Sum[Floor[Floor[n/((b + 1) b^(k - 1) - 1)]/(Floor[n/((b + 1) b^(k - 1) - 1)] - 1/b)] - Floor[Floor[n/(2 b^k - 1)]/(Floor[n/(2 b^k - 1)] - 1/ b)], {k, Floor[Log[b, n]]}]}, Mod[q, b] (b + 1)^c * b^Floor[Log[b, q]] + Sum[Floor[Mod[q, b^(k + 1)]/b^k] b^(Floor[Log[b, q]] - k) (b^(2 k + c) + 1), {k, Floor[Log[b, q]]}]] (* after the work of Eric A. Schmidt, works for all integer bases b > 2 *)
Array[nthPalindromeBase[#, 10] &, 61, 0] (* please note that Schmidt uses a different, a more natural and intuitive offset, that of a(1) = 1. - Robert G. Wilson v, Sep 22 2014 and modified Nov 28 2014 *)
PROG
(PARI) is_ A002113(n)=Vecrev(n=digits(n))==n \\ M. F. Hasler, Nov 17 2008, updated Apr 26 2014, Jun 19 2018 (PARI) is(n)=n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2013 (PARI) a(n)={my(d, i, r); r=vector(#digits(n-10^(#digits(n\11)))+#digits(n\11)); n=n-10^(#digits(n\11)); d=digits(n); for(i=1, #d, r[i]=d[i]; r[#r+1-i]=d[i]); sum(i=1, #r, 10^(#r-i)*r[i])} \\ David A. Corneth, Jun 06 2014 (PARI) \\ recursive--feed an element a(n) and it gives a(n+1)
nxt(n)=my(d=digits(n)); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); sum(i=1, #d, 10^(#d-i)*d[i]) \\ David A. Corneth, Jun 06 2014 (PARI) \\ feed a(n), returns n.
inv(n)={my(d=digits(n)); q=ceil(#d/2); sum(i=1, q, 10^(q-i)*d[i])+10^floor(#d/2)} \\ David A. Corneth, Jun 18 2014 (PARI) inv_ A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P} \\ index n of palindrome P = a(n), much faster than above: no sum is needed. - M. F. Hasler, Sep 09 2018 (PARI) A002113(n, L=logint(n, 10))=(n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(n<L, n, n\10)))) \\ M. F. Hasler, Sep 11 2018 mlist=[]
for n in range(nMax+1):
mstr=str(n)
if mstr==mstr[::-1]:
mlist.append(n)
(Python)
from itertools import chain
A002113 = sorted(chain(map(lambda x:int(str(x)+str(x)[::-1]), range(1, 10**3)), map(lambda x:int(str(x)+str(x)[-2::-1]), range(10**3)))) # Chai Wah Wu, Aug 09 2014
(Python)
from itertools import chain, count
A002113 = chain(k for k in count(0) if str(k) == str(k)[::-1])
(Python)
from math import log10
if n < 2: return 0
P = 10**floor(log10(n//2)); M = 11*P
s = str(n - (P if n < M else M-P))
return int(s + s[-2 if n < M else -1::-1]) # M. F. Hasler, Jun 06 2024 (Haskell)
a002113 n = a002113_list !! (n-1)
(Haskell)
import Data.List.Ordered (union)
a002113_list = union a056524_list a056525_list -- Reinhard Zumkeller, Jul 29 2015, Dec 28 2011 (Magma) [n: n in [0..600] | Intseq(n, 10) eq Reverse(Intseq(n, 10))]; // Vincenzo Librandi, Nov 03 2014 (SageMath)
[n for n in (0..515) if Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018 (GAP) Filtered([0..550], n->ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019 (Scala) def palQ(n: Int, b: Int = 10): Boolean = n - Integer.parseInt(n.toString.reverse) == 0
CROSSREFS
Palindromes in bases 2 through 11: A006995 and A057148, A014190 and A118594, A014192 and A118595, A029952 and A118596, A029953 and A118597, A029954 and A118598, A029803 and A118599, A029955 and A118600, this sequence, A029956. Also A262065 (base 60), A262069 (subsequence). Minimal number of palindromes that add to n using greedy algorithm: A088601. Minimal number of palindromes that add to n: A261675.
a(n) = n + reversal of digits of n.
+10
84
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
COMMENTS
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
MATHEMATICA
Table[n+FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 100}] (* Harvey P. Dale, Jul 19 2014 *)
Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x) = x + (x with digits reversed).
+10
71
196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996, 3493, 3495, 3583, 3585, 3673, 3675
COMMENTS
196 is conjectured to be smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended this to millions of digits without finding one (see A006960). Also called Lychrel numbers, though the definition of "Lychrel number" varies: Purists only call the "seeds" or "root numbers" Lychrel; the "related" or "extra" numbers (arising in the former's orbit) have been coined "Kin numbers" by Koji Yamashita. There are only 2 "root" Lychrels below 1000 and 3 more below 10000, cf. A088753. - M. F. Hasler, Dec 04 2007 Question: when do numbers in this sequence start to outnumber numbers that are not in the sequence? - J. Lowell, May 15 2014 Answer: according to Doucette's site, 10-digit numbers have 49.61% of Lychrels. So beyond 10 digits, Lychrels start to outnumber non-Lychrels. - Dmitry Kamenetsky, Oct 12 2015 From the current definition it is unclear whether palindromes are excluded from this sequence, cf. A088753 vs A063048. 9999 would be the first palindromic term that will never result in a palindrome when the Reverse-then-add function A056964 is repeatedly applied. - M. F. Hasler, Apr 13 2019
REFERENCES
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
LINKS
DeCode, Lychrel Number, dCode.fr 'toolkit' to solve games, riddles, geocaches, 2020. R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
EXAMPLE
Under the "add reverse" operation, we have:
196 (+ 691) -> 887 (+ 788) -> 1675 (+ 5761) -> 7436 (+ 6347) -> 13783 (+ 38731) -> etc. which apparently never leads to a palindrome.
Similar for 295 (+ 592) -> 887, 394 (+ 493) -> 887, 790 (+ 097) -> 887 and 689 (+ 986) -> 1675, which all merge immediately into the above sequence, and also for the reverse of any of the numbers occurring in these sequences: 493, 592, 691, 788, ...
879 (+ 978) -> 1857 -> 9438 -> 17787 -> 96558 is the only other "root" Lychrel below 1000 which yields a sequence distinct from that of 196. (End)
MATHEMATICA
With[{lim = 10^3}, Select[Range@ 4000, Length@ NestWhileList[# + IntegerReverse@ # &, #, ! PalindromeQ@ # &, 1, lim] == lim + 1 &]] (* Michael De Vlieger, Dec 23 2017 *)
PROG
(PARI) select( {is_ A023108(n, L=exponent(n+1)*5)=while(L--&& n*2!=n+= A004086(n), ); !L}, [1..3999]) \\ with { A004086(n)=fromdigits(Vecrev(digits(n)))}; default value for search limit L chosen according to known records A065199 and indices A065198. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020
CROSSREFS
Cf. A056964 ("reverse and add" operation on which this is based).
Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.
+10
33
196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694
COMMENTS
The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999. Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4). If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.
The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.
It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019 Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.
All numbers in this sequence seem to satisfy the following condition:
d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".
The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)
REFERENCES
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
EXAMPLE
1997 is a term since the trajectory of 1997 (presumably) does not lead to a number which occurs in the trajectory of 196 or of 879 (actually checked for the first 10000 terms of these trajectories). The trajectory of 1997 joins the trajectory of 106 at 97768 (cf. A070796), but 106 is not a term of the present sequence.
MATHEMATICA
limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 14000], (x = NestWhileList[ # + IntegerReverse[#] &, #, ! PalindromeQ[#] &, 1, limit];
If[Length[x] >= limit && Intersection[x, utraj] == {},
utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)
Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.
+10
11
1, 2, 2, 4, 4, 3, 8, 8, 6, 4, 16, 16, 12, 8, 5, 77, 77, 33, 16, 10, 6, 154, 154, 66, 77, 11, 12, 7, 605, 605, 132, 154, 22, 33, 14, 8, 1111, 1111, 363, 605, 44, 66, 55, 16, 9, 2222, 2222, 726, 1111, 88, 132, 110, 77, 18, 10, 4444, 4444, 1353, 2222, 176, 363, 121, 154, 99, 11, 11
EXAMPLE
T(5,6) = 88, since 88 is the 6th term in the reverse and add sequence of 5.
Table starts with:
1 2 4 8 16 77 154 605 1111 2222
2 4 8 16 77 154 605 1111 2222 4444
3 6 12 33 66 132 363 726 1353 4884
4 8 16 77 154 605 1111 2222 4444 8888
5 10 11 22 44 88 176 847 1595 7546
6 12 33 66 132 363 726 1353 4884 9768
7 14 55 110 121 242 484 968 1837 9218
8 16 77 154 605 1111 2222 4444 8888 17776
9 18 99 198 1089 10890 20691 40293 79497 158994
10 11 22 44 88 176 847 1595 7546 14003
MAPLE
T:= proc(n, r) option remember; `if`(r=1, n, (h-> h +(s->
parse(cat(s[-i]$i=1..length(s))))(""||h))(T(n, r-1)))
end:
MATHEMATICA
rad[n_] := n + FromDigits[Reverse[IntegerDigits[n]]];
T[n_, 1] := n; T[n_, k_] := T[n, k] = rad[T[n, k-1]];
a(n) = smallest non-palindromic number k such that the Reverse and Add! trajectory of k joins the trajectory of A089521(n).
+10
1
19098, 199998, 990999, 1909098, 19002998, 9905999, 11009094, 19003098, 19005098, 19007098, 19999998, 190009098, 990959999, 990969999, 1990029998, 1990049998, 1990069998, 1990089998, 999079999, 1999049998, 1999069998
COMMENTS
a(12), a(15) to a(18), a(20), a(21) are conjectural; it is not yet ensured that they are minimal.
EXAMPLE
The trajectories of A088753(1) = 9999 and of 19098 join at 2089791 and there is no number between 9999 and 19098 whose trajectory joins that of 9999.
Primes in A023108(n); or Lychrel primes.
+10
0
691, 887, 1997, 3583, 3673, 3853, 3943, 4079, 4259, 4349, 4799, 4889, 5581, 5851, 6257, 6977, 8089, 8179, 8269, 8539, 8629, 8719, 10663, 10883, 11777, 11833, 11867, 11923, 11953, 11959, 12097, 12763, 12823, 13397, 13523, 13553, 13597, 13633
COMMENTS
A023108(n) = Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits reversed).
EXAMPLE
A023108(n) begins {196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, ...}.
Thus a(1) = 691 which is the first prime in A023108(n), a(2) = 887 which is the second prime in A023108(n).
Search completed in 0.011 seconds
| 
