A004086 -id:A004086 - cp's OEIS frontend

A068634 a(n) = lcm(n, R(n)), where R(n) (A004086) = digit reversal of n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 84, 403, 574, 255, 976, 1207, 162, 1729, 20, 84, 22, 736, 168, 1300, 806, 216, 1148, 2668, 30, 403, 736, 33, 1462, 1855, 252, 2701, 3154, 1209, 40, 574, 168, 1462, 44, 270, 1472, 3478, 336, 4606, 50, 255, 1300, 1855, 270, 55

Offset: 1

Amarnath Murthy, Feb 27 2002

			a(12) = lcm(12,21) = 84.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

r:= proc(n) local q;
      `if`(n<10, n, irem(n, 10, 'q') *10^(length(n)-1) +r(q))
    end:
a:= n-> ilcm(n, r(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 16 2012

Array[LCM[#,FromDigits[Reverse[IntegerDigits[#]]]]&,50] (* Harvey P. Dale, Dec 13 2011 *)

More terms from Alois P. Heinz, Aug 16 2012

A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

Original entry on oeis.org

100, 144, 169, 200, 288, 300, 400, 441, 500, 528, 600, 700, 768, 800, 825, 867, 882, 900, 961, 1089, 1100, 1584, 2178, 2200, 3300, 4400, 4851, 5500, 6600, 7700, 8712, 8800, 9801, 9900, 10000, 10100, 10404, 10609, 10989

Offset: 1

Reinhard Zumkeller, May 15 2002

If k is a palindrome (A002113), then 100*k is a term. If k is a term, then 100*k is a term. - Chai Wah Wu, Mar 31 2018
From Bernard Schott, Jan 02-10 2019: (Start)
There are six different families of integers in this sequence.
1) If k and rev(k) do not have the same number of digits:
All these integers are in A322835 where the first four families are explained and detailed.
Family 1: A002113(j) * 100^k
Family 2: A035090(j) * 100^k
Family 3: A082994(j) * 100^k
Family 4: A323061(j) * 10^(2k+1)
2) If k and rev(k) have the same number of digits.
All these integers are in A062917.
Family 5: Non-palindromic squares whose reverse is also square. These integers are in A035090.
Family 6: Non-palindromic numbers k, such that k * rev(k) is a square, with k and rev(k) not both square. These integers are in A082994.
3) Relationships between these different sequences.
  A035090 Union A082994 = A062917 with empty intersection, and,
A062917 Union A322835 = {This sequence} with empty intersection. (End)

			a(2)=144: rev(144)=441, 144*441=(12^2)*(21^2)=(12*21)^2 and 144<>12*21=252.
From _Bernard Schott_, Jan 02 2019: (Start)
Example for family 1: 200 * 2 = 400 = 20^2
Example for family 2: 14400 * 441 = 120^2 * 21^2 = 2520^2
Example for family 3: 28800 * 882 = (2 * 120^2) * (2 * 21^2) = 5040^2
Example for family 4: 5449680 * 869445 = 2176740^2
Example for family 5: 169 * 961 = 13^2 * 31^2 = 403^2
Example for family 6: 528 * 825 = (33 * 4^2) * (33 * 5^2) = 660^2. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Reinhard Zumkeller)

Cf. A002113, A061205, A010052, A035090, A062917, A082994, A322835, A323061.

a070760 n = a070760_list !! (n-1)
a070760_list = [x | x <- [0..], let y = a061205 x,
                    y /= x ^ 2, a010052 y == 1]
-- Reinhard Zumkeller, Apr 10 2012, Apr 29 2011

Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* Jean-François Alcover, Nov 30 2011 *)
sdnQ[n_]:=Module[{c=n*IntegerReverse[n]},c!=n^2&&IntegerQ[Sqrt[c]]]; Select[ Range[11000],sdnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2016 *)

A102118 Iccanobirt numbers (8 of 15): a(n) = R(a(n-1) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 31, 24, 26, 18, 86, 31, 531, 846, 8041, 8149, 63071, 16297, 71578, 649051, 629637, 6620531, 9129987, 55108361, 97885807, 551421261, 924514407, 5741283751, 9149127127, 58252941851, 92725334137, 511304721061

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 30 2004

Digit reversal variation of tribonacci numbers A000073.

Inspired by Iccanobif numbers: A001129, A014258-A014260.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A102111, A102112, A102113, A102114, A102115, A102116, A102117, A102119, A102120, A102121, A102122, A102123, A102124, A102125.

R[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Clear[a];a[0]=0;a[1]=0;a[2]=1;a [n_]:=a[n]=R[a[n-1]+a[n-2]+a[n-3]];Table[a[n], {n, 0, 40}]
Transpose[NestList[{#[[2]],#[[3]],FromDigits[Reverse[IntegerDigits[Total[ #]]]]}&,{0,0,1},40]][[1]] (* Harvey P. Dale, Dec 04 2012 *)

Incorrect formula removed by Georg Fischer, Dec 18 2020

A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 101, 111, 121, 131, 141, 144, 151, 161, 169, 171, 181, 191, 200, 202, 212, 222, 232, 242, 252, 262, 272, 282, 288, 292, 300, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 400, 404, 414, 424, 434, 441, 444, 454, 464, 474, 484, 494, 500, 505, 515, 525, 528, 535

Offset: 1

Bernard Schott, Feb 02 2019

The first nineteen terms are palindromes (cf. A002113). There are exactly seven different families of integers which together partition the terms of this sequence. See the file "Sequences and families" for more details, comments, formulas and examples.
From Chai Wah Wu, Feb 18 2019: (Start)
If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 where k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then n*R(n) = w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.
The same argument shows that numbers corresponding to axaxa, axaxaxa, ... are also terms.
For example, since 528 is a term, so are 528528, 5280528, 52800528, 5280052800528, etc.
(End)

			One example for each family:
family 1 is A002113: 323 * 323 = 323^2;
family 2 is A035090: 169 * 961 = 13^2 * 31^2 = 403^2;
family 3 is A082994: 288 * 882 = (2*144) * (2*441) = 504^2;
family 4 is A002113(j) * 100^k: 75700 * 757 = 7570^2;
family 5 is A035090(j) * 100^k: 44100 * 144 = 2520^2;
family 6 is A082994(j) * 100^k: 8670000 * 768 = 81600^2;
family 7 is A323061(j) * 10^(2k+1): 5476580 * 856745 = 2166110^2.

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), p. 168.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernard Schott, Sequences and Families
Eric Weisstein's World of Mathematics, Reversal

Cf. A002113, A070760, A062917, A035090, A082994, A322835, A323061.

Cf. A083406, A083407, A083408, A117281 (Squares = k * rev(k) in at least two ways).

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= n -> issqr(n*revdigs(n)):
select(filter, [$0..1000]);# Robert Israel, Feb 09 2019

Select[Range[0, 535], IntegerQ@ Sqrt[# IntegerReverse@ #] &] (* Michael De Vlieger, Feb 03 2019 *)

isok(n) = issquare(n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Feb 04 2019

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

Original entry on oeis.org

18, 81, 198, 576, 675, 819, 891, 918, 1131, 1304, 1311, 1818, 1998, 2262, 2622, 3393, 3933, 4031, 4154, 4514, 4636, 6364, 8181, 8749, 8991, 9478, 12441, 14269, 14344, 14421, 15167, 15602, 16237, 18018, 18449, 18977, 19998, 20651, 23843, 24882, 26677, 26892, 27225

Offset: 1

Michel Marcus, Oct 08 2020

Palindromes (A002113) are excluded from the sequence because they obviously satisfy the condition.
Sequence is infinite since it includes 18, 1818, 181818, .... See link.
There are many cases of terms that are the repeated concatenation of integers like: 1818, 8181, 181818, ... , but also 131313131313131313131313131313 and more. See A338166.
If n is in the sequence and has d digits, and gcd(n, x) = gcd(A004086(n), x) where x = (10^((k+1)*d)-1)/(10^d-1), then the concatenation of k copies of n is also in the sequence. - Robert Israel, Oct 13 2020

			For m = 18 = 2*3^2, A338038(18) = 2 + (3+2) = 7 and for m = 81 = 3^4, A338038(81) = 7, so 18 and 81 are terms.

Michel Marcus, Table of n, a(n) for n = 1..2998
Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Muhammet Boran, Garam Choi, Steven J. Miller, Jesse Purice, and Daniel Tsai, A characterization of prime v-palindromes, arXiv:2307.00770 [math.NT], 2023.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.
Daniel Tsai, v-palindromes: an analogy to the palindromes, arXiv:2111.10211 [math.HO], 2021.

Cf. A004086 (read n backwards), A002113, A029742 (non-palindromes), A338038, A338166.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
g:= proc(n) local t;
  add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]))
end proc:
filter:= proc(n) local r;
  if n mod 10 = 0 then return false fi;
  r:= rev(n);
  r <> n and g(r)=g(n)
end proc:
select(filter, [$1..30000]); # Robert Israel, Oct 13 2020

s[1] = 0; s[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Select[Range[30000], !Divisible[#, 10] && (r = IntegerReverse[#]) != # &&  s[#] == s[r] &] (* Amiram Eldar, Oct 08 2020 *)

f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
isok(m) = my(r=fromdigits(Vecrev(digits(m)))); (m % 10) && (m != r) && (f(r) == f(m));

A068637 a(n) = Max(n, R(n)), where R(n) (A004086) = digit reversal of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 31, 41, 51, 61, 71, 81, 91, 20, 21, 22, 32, 42, 52, 62, 72, 82, 92, 30, 31, 32, 33, 43, 53, 63, 73, 83, 93, 40, 41, 42, 43, 44, 54, 64, 74, 84, 94, 50, 51, 52, 53, 54, 55, 65, 75, 85, 95, 60, 61, 62, 63, 64, 65, 66, 76, 86, 96, 70, 71

Offset: 1

Amarnath Murthy, Feb 27 2002

a(n) = A004186(n) for n <= 100. - Reinhard Zumkeller, Apr 03 2015

			a(12) = max(12,21) = 21. a(34632) = max(34632,23643) = 34632.

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

Cf. A068636.

Cf. A004086, A004186.

a068637 n = max n $ a004086 n  -- Reinhard Zumkeller, Apr 03 2015

a:= n-> max(n,(s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2015

Table[Max[n,IntegerReverse[n]],{n,100}] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jan 10 2016 *)

def A068637(n): return max(n,int(str(n)[::-1])) # Chai Wah Wu, Jun 26 2025

Wrong a(13)=22 removed by Reinhard Zumkeller, Apr 03 2015

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

A071249 Numbers k such that gcd(k, R(k)) > 1, where R(k) (A004086) is the digit reversal of k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 33, 36, 39, 40, 42, 44, 45, 46, 48, 50, 51, 54, 55, 57, 60, 62, 63, 64, 66, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 93, 96, 99, 101, 102, 105, 108, 110, 111, 114, 117, 120, 121

Offset: 1

Amarnath Murthy, May 21 2002

Numbers k such that A055483(k) > 1. - Reinhard Zumkeller, Jun 18 2013

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

Cf. A004086, A055483, A226778 (complement).

a071249 n = a071249_list !! (n-1)
a071249_list = filter ((> 1) . a055483) [1..]
-- Reinhard Zumkeller, Jun 18 2013

Select[ Range[125], GCD[ #, FromDigits[ Reverse[ IntegerDigits[ # ]]]] > 1 & ]

Edited by Robert G. Wilson v, Jun 07 2002

Definition corrected by N. J. A. Sloane, Aug 27 2020 following a suggestion from José Hernández Santiago

A071273 Concatenation of R(n) (A004086) and n, omitting leading 0's.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1111, 2112, 3113, 4114, 5115, 6116, 7117, 8118, 9119, 220, 1221, 2222, 3223, 4224, 5225, 6226, 7227, 8228, 9229, 330, 1331, 2332, 3333, 4334, 5335, 6336

Offset: 1

Amarnath Murthy, Jun 07 2002

			a(12) = 2112, a(1235) = 53211235

Cf. A071274.

Table[10^IntegerLength[n] IntegerReverse[n]+n,{n,40}] (* Harvey P. Dale, Mar 14 2023 *)

A072016 Numbers k such that gcd(k, reverse(k)) = 27 = 3^3, where reverse(x) = A004086(x).

Original entry on oeis.org

2889, 3699, 3888, 3969, 4779, 4887, 5589, 5697, 5778, 5859, 5886, 5967, 6399, 6669, 6777, 6885, 6939, 7398, 7479, 7587, 7668, 7695, 7749, 7776, 7857, 7884, 7938, 7965, 8289, 8397, 8559, 8667, 8775, 8829, 8883, 8937, 9099, 9288, 9369, 9396, 9477, 9558, 9585

Offset: 1

Labos Elemer, Jun 05 2002

Solutions to gcd(k, reverse(k)) = 1,3,9 (lower powers of 3) are trivial (see A072005).

			2889 = 107*3*3*3, 9889 = 3*3*3*3*2*61.

Vincenzo Librandi, Table of n, a(n) for n = 1..1030

Cf. A004086, A055483, A069554, A071686, A072005.

[n: n in [1..10^4] | Gcd(n,Seqint(Reverse(Intseq(n)))) eq 27]; // Vincenzo Librandi, Jul 11 2018

Select[Range[10^4], GCD[#, FromDigits[Reverse[IntegerDigits[#]]]] == 27 &] (* Vincenzo Librandi, Jul 11 2018 *)

isok(n) = gcd(n, fromdigits(Vecrev(digits(n)))) == 27; \\ Michel Marcus, Jul 11 2018

cp's OEIS Frontend

A068634 a(n) = lcm(n, R(n)), where R(n) (A004086) = digit reversal of n.

Views

Author

Keywords

Examples

Links

Programs

Maple

Mathematica

Extensions

A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A102118 Iccanobirt numbers (8 of 15): a(n) = R(a(n-1) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A306273 Numbers k such that k * rev(k) is a square, where rev=A004086, decimal reversal.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A338039 Numbers m such that A338038(m) = A338038(A004086(m)) where A004086(i) is i read backwards and A338038(i) is the sum of the primes and exponents in the prime factorization of i ignoring 1-exponents; palindromes and multiples of 10 are excluded.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A068637 a(n) = Max(n, R(n)), where R(n) (A004086) = digit reversal of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Extensions

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python