A004086 -id:A004086 - cp's OEIS frontend

A072050 Smallest solution to GCD(x,A004086(x))=7^n.

7, 18718, 343, 125204947, 231012215, 11298657013, 211066659013, 117088913464607, 2846847905744815, 108244538579770418, 2080795357577501075, 18312871825384462928, 26268977180287044053417, 1734582041294009627423816

Offset: 1

Labos Elemer, Jun 10 2002

Cf. A004086, A055483, A069554, A071686, A072005, A072021.

Mathematica
```
(* See A071686. *)
```

a(n) = A069554(7^n).

a(8)-a(9) from Max Alekseyev, Jun 17 2011

a(10)-a(14) from Giovanni Resta, Oct 30 2019

A100413 Numbers k such that k is reversal(k)-th even composite number (k is A004086(k)-th even composite number).

Original entry on oeis.org

52, 592, 5992, 59992, 599992, 5999992, 59999992, 599999992, 5999999992, 59999999992, 599999999992, 5999999999992, 59999999999992, 599999999999992, 5999999999999992, 59999999999999992, 599999999999999992

Offset: 1

Farideh Firoozbakht, Dec 08 2004

			592 is in the sequence because 592 is the 295th even composite number.

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A004086, A086943, A100412.

[6*10^n -8: n in [1..20]]; // G. C. Greubel, Apr 13 2023

A100413:=n->6*10^n-8; seq(A100413(n), n=1..20); # Wesley Ivan Hurt, Apr 06 2014

Mathematica
```
Table[6*10^n-8, {n,20}]
```

A100413(n):=6*10^n-8$
makelist(A100413(n),n,1,17); /* Martin Ettl, Nov 08 2012 */

Vec(4*x*(5*x+13)/((x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Oct 14 2014

[6*10^n -8 for n in range(1,21)] # G. C. Greubel, Apr 13 2023

a(n) = 6*10^n - 8.
a(n) = 2*(A086943(n) + 3). - Martin Ettl, Nov 08 2012
From Colin Barker, Oct 14 2014: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3).
G.f.: 4*x*(13+5*x)/((1-x)*(1-10*x)). (End)
E.g.f.: 2 (1 - 4*exp(x) + 3*exp(10*x)). - G. C. Greubel, Apr 13 2023

A323061 Numbers n that are not multiples of 10 and such that 10nR(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.

Original entry on oeis.org

544968, 547658, 560106, 601065, 856745, 869445, 2495295, 4601685, 5606106, 5861064, 5925942, 6016065, 20861005, 21778875, 22972005, 29389855, 42251835, 50016802, 50027922, 51826326, 53815224, 55898392, 56066106, 56570706, 56873466, 57887712, 60166065, 60707565

Offset: 1

Chai Wah Wu, Jan 07 2019

10*a(n) are exactly the terms in A322835 that are not multiples of 100.
m is a term if and only if R(m) is a term.
The product of the first and last digits of a term is a multiple of 10, i.e., the first and last digits of a term are the digit 5 and an even nonzero digit.
The sequence has an infinite number of terms. For instance, 601x065 is a term where x is a string of k repeated digits 6 and k >= 0, i.e., 601065, 6016065, 60166065, etc. Similarly numbers of the form 560x106 are also terms.
To see this, let a = 601*10^(3+k) + 65 + 6000*(10^k-1)/9. Then R(a) = 56*10^(4+k) + 106 + 6000*(10^k-1)/9. The number 10*a*R(a) can be written as 30360100*(10^(k + 3) - 1)^2/9 whose square root is 5510*(10^(k + 3) - 1)/3.
From Chai Wah Wu, Feb 18 2019: (Start)
22994x77005 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 22994*10^(5+k) + 77005 + 100000*(10^k-1). Then R(a) = 50077*10^(5+k) + 49922 + 100000*(10^k-1). The number 10*a*R(a) can be written as 11515436100*(10^(k+5) - 1)^2, whose square root is 107310*(10^(k+5) - 1).
23804x76195 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 23804*10^(5+k) + 76195 + 100000*(10^k-1). Then R(a) = 59167*10^(5+k) + 40832 + 100000*(10^k-1). The number 10*a*R(a) can be written as 14084942400*(10^(k+5) - 1)^2, whose square root is 118680*(10^(k+5) - 1).
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 and k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Note that n is also not a multiple of 10. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then 10*n*R(n) = 10*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 544968 is a term, so are 544968544968, 5449680544968, 54496800544968, 5449680054496800544968, etc.
(End)

			238026195 * 591620832 * 10 = 1186681320^2.

Giovanni Resta, Table of n, a(n) for n = 1..200 (first 54 terms from Chai Wah Wu)

Cf. A004086, A322835.

Select[Range[61*10^6],Mod[#,10]!=0&&IntegerQ[Sqrt[10# IntegerReverse[ #]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 16 2020 *)

isok(n) = (n % 10) && issquare(10*n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Jan 10 2019

A068636 a(n) = Min(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, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 2, 12, 22, 23, 24, 25, 26, 27, 28, 29, 3, 13, 23, 33, 34, 35, 36, 37, 38, 39, 4, 14, 24, 34, 44, 45, 46, 47, 48, 49, 5, 15, 25, 35, 45, 55, 56, 57, 58, 59, 6, 16, 26, 36, 46, 56, 66, 67, 68, 69, 7, 17, 27, 37, 47, 57

Offset: 1

Amarnath Murthy, Feb 27 2002

a(n) = A004185(n) for n <= 100. - Reinhard Zumkeller, Apr 03 2015

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

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

Cf. A068637, A004086, A004185.

a068636 n = min n $ a004086 n  -- Reinhard Zumkeller, Apr 03 2015

a:= n-> min(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[Min[n,IntegerReverse[n]],{n,80}] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Nov 27 2015 *)

def A068636(n): return min(n,int(str(n)[::-1])) # Chai Wah Wu, Jun 26 2025

A071241 Arithmetic mean of k and R(k) where k is the n-th nonnegative number using only even digits and R(k) is its digit reversal (A004086).

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 22, 33, 44, 55, 66, 33, 44, 55, 66, 77, 44, 55, 66, 77, 88, 101, 202, 303, 404, 505, 121, 222, 323, 424, 525, 141, 242, 343, 444, 545, 161, 262, 363, 464, 565, 181, 282, 383, 484, 585, 202, 303, 404, 505, 606, 222, 323, 424, 525

Offset: 0

Amarnath Murthy, May 20 2002

Conjecture: 101 is the largest prime term, the only other primes being 2 and 11.

The conjecture is false: for example, 181 and 383 are prime terms. There are 150 prime terms less than 75000. - Harvey P. Dale, Sep 02 2016

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Cf. A004086, A014263, A071240, A071242.

reversal := proc(n) local i, len, new, temp: new := 0: temp := n: len := floor(log[10](n+.1))+1: for i from 1 to len do new := new+irem(temp, 10)*10^(len-i): temp := floor(temp/10): od: RETURN(new): end: alleven := proc(n) local i, flag, len, temp: temp := n: flag := 1: if n=0 then flag := 0 fi: len := floor(log[10](n+.1))+1: for i from 1 to len do if irem(temp, 10) mod 2 = 0 then temp := floor(temp/10) else flag := 0 fi: od: RETURN(flag): end: for n from 0 to 500 by 2 do if alleven(n) = 1 then printf(`%d,`,(n+reversal(n))/2) fi: od: # James Sellers, May 28 2002

Mean[{#,IntegerReverse[#]}]&/@(FromDigits/@Tuples[{0,2,4,6,8},3]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 02 2016 *)

{k + R(k)}/2 where k uses only odd digits 0, 2, 4, 6 and 8.

a(n) = (A014263(n) + A004086(A014263(n))) / 2. - Sean A. Irvine, Jul 06 2024

More terms from James Sellers, May 28 2002

Corrected by Harvey P. Dale, Sep 02 2016

A081431 RevBinary(RevDecimal(n)), where RevBinary(m) is the binary reversal of m (A030101) and RevDecimal(m) is the decimal reversal of m (A004086).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 1, 13, 21, 31, 37, 51, 47, 113, 69, 109, 1, 3, 13, 1, 21, 11, 31, 9, 37, 29, 3, 11, 29, 33, 53, 43, 63, 73, 101, 93, 1, 7, 3, 17, 13, 27, 1, 41, 21, 61, 5, 15, 19, 49, 45, 59, 65, 105, 85, 125, 3, 1, 11, 9, 29, 7, 33, 25, 53, 3, 7, 17, 27, 41

Offset: 0

Reinhard Zumkeller, Mar 20 2003

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

Cf. A004086, A030101.

Cf. A081432, A081433, A007088.

Table[IntegerReverse[IntegerReverse[n],2],{n,0,80}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 03 2018 *)

a(n) = fromdigits(Vecrev(binary(fromdigits(Vecrev(digits(n))))), 2); \\ Michel Marcus, Jan 30 2023

def a(n): return int(bin(int(str(n)[::-1]))[:1:-1], 2)
print([a(n) for n in range(74)]) # Michael S. Branicky, Jan 30 2023

a(n) = A030101(A004086(n)). - Michel Marcus, Jan 30 2023

A081432 RevDecimal(RevBinary(n)), where RevDecimal(m) is the decimal reversal of m (A004086) and RevBinary(m) is the binary reversal of m (A030101).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 31, 3, 11, 7, 51, 1, 71, 9, 52, 5, 12, 31, 92, 3, 91, 11, 72, 7, 32, 51, 13, 1, 33, 71, 94, 9, 14, 52, 75, 5, 73, 12, 35, 31, 54, 92, 16, 3, 53, 91, 15, 11, 34, 72, 95, 7, 93, 32, 55, 51, 74, 13, 36, 1, 56, 33, 79, 71, 18, 94, 311, 9, 37, 14

Offset: 0

Reinhard Zumkeller, Mar 20 2003

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

Cf. A030101, A004086.

Cf. A081431, A081433, A007088.

IntegerReverse[IntegerReverse[Range[0,80],2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 03 2017 *)

a(n) = fromdigits(Vecrev(digits(fromdigits(Vecrev(binary(n)), 2)))); \\ Michel Marcus, Jan 30 2023

def a(n): return int(str(int(bin(n)[:1:-1], 2))[::-1])
print([a(n) for n in range(75)]) # Michael S. Branicky, Jan 30 2023

a(n) = A004086(A030101(n)). - Michel Marcus, Jan 30 2023

A081433 Numbers n such that RevBinary(RevDecimal(n))=RevDecimal(RevBinary(n)), where RevDecimal(n) is the decimal reversal of n (A004086) and RevBinary(n) is the binary reversal of n (A030101).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 33, 51, 66, 99, 165, 313, 516, 535, 561, 585, 615, 626, 717, 759, 858, 929, 957, 1241, 1421, 2112, 2482, 2552, 2842, 3579, 4224, 5485, 5845, 6336, 7447, 8448, 9009, 9753, 11051, 12631, 13621, 15011, 15351, 15375

Offset: 1

Reinhard Zumkeller, Mar 20 2003

A030101(A004086(a(n)))=A004086(A030101(a(n))), A081431(a(n))=A081432(a(n)).

Cf. A081432, A081434, A007088.

def ok(n): return int(bin(int(str(n)[::-1]))[:1:-1], 2) == int(str(int(bin(n)[:1:-1], 2))[::-1])
print([k for k in range(2**14) if ok(k)]) # Michael S. Branicky, Jan 30 2023

A060568 Number of primes between n and R(n) where R(n) (A004086) is the digit reversal of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 3, 4, 6, 9, 11, 12, 15, 16, 7, 3, 0, 2, 4, 6, 9, 11, 13, 14, 8, 4, 2, 0, 2, 4, 7, 8, 10, 12, 10, 6, 4, 2, 0, 2, 4, 6, 8, 9, 12, 9, 6, 4, 2, 0, 2, 5, 7, 7, 14, 11, 9, 7, 4, 2, 0, 2, 4, 5, 15, 12, 11, 8, 6, 5, 2, 0, 2, 2, 18, 15, 13, 10, 8, 7, 4, 2, 0, 1, 20, 16, 14

Offset: 1

Amarnath Murthy, Apr 27 2001

			a(10) = 4, as there are four primes between 10 and 1.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A000720, A074813.

from sympy import primerange
def A060568(n):
    x = (n, int(str(n)[::-1],10))
    return len([i for i in primerange(min(x)+1,max(x))]) # John Tyler Rascoe, Jan 13 2025

More terms from Larry Reeves (larryr(AT)acm.org), May 10 2001

A062936 Numbers n such that n*R(n) is a palindrome, where R(n) (A004086) = digit reversal.

Original entry on oeis.org

1, 2, 3, 11, 12, 21, 22, 101, 102, 111, 112, 121, 122, 201, 202, 211, 212, 221, 1001, 1002, 1011, 1012, 1021, 1022, 1101, 1102, 1111, 1112, 1121, 1201, 1202, 1211, 2001, 2002, 2011, 2012, 2021, 2101, 2102, 2111, 2201, 10001, 10002, 10011, 10012

Offset: 1

Amarnath Murthy, Jul 05 2001

			122*221 = 26962 hence 122 belongs to the sequence.

Harry J. Smith and Indranil Ghosh, Table of n, a(n) for n = 1..4357 (first 500 terms from Harry J. Smith)
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.

Cf. A048343, A048344.

Select[Range[100000], Reverse[IntegerDigits[ #*FromDigits[Reverse[IntegerDigits[ # ]]]]] == IntegerDigits[ #*FromDigits[Reverse[IntegerDigits[ # ]]]] &] (* Tanya Khovanova, Jun 17 2009 *)
Select[Range[11000],PalindromeQ[# IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 21 2020 *)

lista(nn) = for(n=1, nn, my(d=digits(n*eval(concat(Vecrev(Str(n)))), 10)); if(d == Vecrev(d), print1(n, ", "))); \\ Altug Alkan, Mar 26 2016

A062936_list = []
for n in range(1,10**5):
    s = str(n*int(str(n)[::-1]))
    if s == s[::-1]:
        A062936_list.append(n) # Chai Wah Wu, Sep 08 2014

Includes integers not ending in 0 with sum of squares of digits < 10. - David W. Wilson, Jul 06 2001

Corrected and extended by Dean Hickerson and Patrick De Geest, Jul 06 2001

cp's OEIS Frontend

A072050 Smallest solution to GCD(x,A004086(x))=7^n.

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A100413 Numbers k such that k is reversal(k)-th even composite number (k is A004086(k)-th even composite number).

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A323061 Numbers n that are not multiples of 10 and such that 10*n*R(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.

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A068636 a(n) = Min(n, R(n)), where R(n) (A004086) = digit reversal of n.

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A071241 Arithmetic mean of k and R(k) where k is the n-th nonnegative number using only even digits and R(k) is its digit reversal (A004086).

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Maple

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A081431 RevBinary(RevDecimal(n)), where RevBinary(m) is the binary reversal of m (A030101) and RevDecimal(m) is the decimal reversal of m (A004086).

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A081432 RevDecimal(RevBinary(n)), where RevDecimal(m) is the decimal reversal of m (A004086) and RevBinary(m) is the binary reversal of m (A030101).

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Mathematica

A323061 Numbers n that are not multiples of 10 and such that 10nR(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.