A055483 -id:A055483 - cp's OEIS frontend

A072018 Numbers k for which gcd(k, reverse(k)) = 243 = 3^5.

4899999987, 4989999897, 4999889997, 4999997889, 5889998997, 5889999969, 5898989997, 5898998988, 5899899789, 5899979979, 5899987998, 5899989699, 5899996989, 5979999879, 5988899997, 5988998898, 5989889979, 5989897998

Offset: 1

Labos Elemer, Jun 05 2002

			k = 4899999987 = 3*3*3*3*3*157*128437 and reverse(k) = 78999999984 = 2*2*2*2*3*3*3*3*3*3*2031893, gcd = 243. Numerous but not all solutions are obtained by inserting strings of 9's between digits of A071016. Further such regular transformations exist.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A004086, A055483, A069554, A071686, A072005, A072016.

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)).

A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

Original entry on oeis.org

1089, 2178, 8712, 9801, 10989, 21978, 87912, 98901, 109989, 219978, 879912, 989901, 1099989, 2199978, 8799912, 9899901, 10891089, 10999989, 21782178, 21999978, 87128712, 87999912, 98019801, 98999901, 108901089, 109999989

Offset: 1

Labos Elemer, Jun 03 2002

The quotient R(n)/n or n/R(n) is always 4 or 9.
This is the union of the four sequence A001232, A222814, A008918, A222815. Equivalently, the union of A008919 and A031877.
There are 4*Fibonacci(floor((n-2)/2)) terms with n digits (this is 2*A214927 or essentially 4*A103609). - Ray Chandler, Oct 12 2017
Conjecture: every term mod 100 is equal to 1, 12, 78, or 89. - Harvey P. Dale, Dec 13 2017

			Palindromic solutions like 12021 or also solutions divisible by 10 were filtered out like {8380,838; q=10} or {8400,48; q=175}. In case of m>R(m), q=m/R(m)=4 or 9.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A001232, A222814, A008918, A222815, A008919, A031877.

Cf. A004086, A055483, A069554, A103609, A214927.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[IntegerDigits[x]] Do[s=Mod[Max[{n, tn[red[n]]}], Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[Mod[n, 10], 0] &&!Equal[n, r], Print[{n, r/n}]], {n, 1, 1000000}]
npnQ[n_]:=Module[{r=IntegerReverse[n]},!PalindromeQ[n]&&!Divisible[ n,10] &&(Mod[n,r]==0||Mod[r,n]==0)]; Select[Range[11*10^7],npnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

x = q*R(x), q is an integer q<>1, q<>10^j and neither of x or R(x) is divisible by 10.

Corrected and extended by Harvey P. Dale, Jul 01 2013
Edited by N. J. A. Sloane, Jul 02 2013
Missing terms inserted by Ray Chandler, Oct 09 2017
Incorrect comment removed by Ray Chandler, Oct 12 2017

A071687 Non-palindromic numbers such that either x=q1.Rev[x] or Rev[x]=q2.x, where R[x]=A004086[x] and q1 or q2 are integers not divisible by 10.

Original entry on oeis.org

510, 540, 810, 1089, 2100, 2178, 4200, 5200, 5610, 5700, 5940, 6300, 8400, 8712, 8910, 9801, 10989, 21978, 23100, 27000, 46200, 51510, 52200, 52800, 54540, 56610, 57200, 59940, 65340, 69300, 81810, 87912, 89910, 98901, 109989, 212100, 217800

Offset: 1

Labos Elemer, Jun 03 2002

			Includes special cases of A071685. Examples represented by {n, Rev[n], integer-quotient} triples: {1089, 9801, 9}, {87912, 21979, 4}, {5610, 165, 34}, {610000, 16, 38125}, etc.

Cf. A071685, A004046, A055483, A069554.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[ed[x]] Do[s=Mod[ma=Max[{n, tn[red[n]]}], mi=Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[n, r] &&!Equal[Mod[ma/mi, 10], 0], Print[{n, r, Max[r/n, n/r]}]], {n, 1, 1000000}]

q=Max[n/Rev[n], Rev[n]/n]=10m+r integer, where r>0, q>1.

A069652 GCD of all the numbers obtained by permuting the digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 3, 1, 1, 3, 1, 1, 9, 1, 2, 3, 22, 1, 6, 1, 2, 9, 2, 1, 3, 1, 1, 33, 1, 1, 9, 1, 1, 3, 4, 1, 6, 1, 44, 9, 2, 1, 12, 1, 5, 3, 1, 1, 9, 55, 1, 3, 1, 1, 6, 1, 2, 9, 2, 1, 66, 1, 2, 3, 7, 1, 9, 1, 1, 3, 1, 77, 3, 1, 8, 9, 2, 1, 12, 1, 2, 3, 88, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 99, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 111

Offset: 1

Amarnath Murthy, Mar 29 2002

			a(84) = gcd(48,84) = 12 = a(48).
a(101) = gcd(101,11) = 1.

Different from A055483.

Edited and extended by Robert G. Wilson v, Apr 02 2002

a(71) corrected by Charles R Greathouse IV, Mar 18 2011

A072017 Numbers k such that gcd(k, reverse(k)) = 81 = 3^4, where reverse(x) = A004086(x).

Original entry on oeis.org

2899999989, 2989999899, 2999889999, 3799999899, 3898989999, 3899799999, 3899999988, 3979989999, 3988899999, 3989999898, 3989999979, 3998999889, 3999889998, 3999898989, 3999899799, 3999979989, 3999988899, 4699998999

Offset: 1

Labos Elemer, Jun 05 2002

Numerous solutions can be constructed by inserting strings of suitable digits between digits of terms in A071016.

			k = 3*3*3*3*3*449*64157 and reverse(k) = 2*2*3*3*3*3*31*67*14827, GCD = 81.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A004086, A055483, A069554, A071686, A072005, A072016, A072018.

A337927 a(n) = n / GCD (n, reverse of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 4, 13, 14, 5, 16, 17, 2, 19, 10, 7, 1, 23, 4, 25, 13, 3, 14, 29, 10, 31, 32, 1, 34, 35, 4, 37, 38, 13, 10, 41, 7, 43, 1, 5, 23, 47, 4, 49, 10, 17, 52, 53, 6, 1, 56, 19, 58, 59, 10, 61, 31, 7, 32, 65, 1, 67, 34, 23, 10, 71, 8, 73, 74, 25, 76, 1, 26, 79

Offset: 1

Ctibor O. Zizka, Oct 05 2020

a(n)=n for n in A226778. - Robert Israel, Oct 09 2020

			a(12) = 4 since 12 / gcd(12,21) = 4. a(101) = 101 / gcd(101,101) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000027, A002113, A004086, A055483, A226778.

rev:= proc(n) local L,k;
  L:= convert(n,base,10);
  add(L[-k]*10^(k-1),k=1..nops(L))
end proc:
f:= n -> n/igcd(n,rev(n)):
map(f, [$1..100]); # Robert Israel, Oct 09 2020

Table[n/GCD[n, IntegerReverse[n]], {n, 1, 100}] (* Amiram Eldar, Oct 05 2020 *)

a(n) = n/gcd(n, fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Oct 06 2020

a(n) = n / gcd(n, A004086(n)).
a(n) = n / A055483(n).
a(n) = A000027(n) / A055483(n).

A071678 GCD of n! and the reverse of n!.

Original entry on oeis.org

1, 2, 6, 6, 3, 9, 45, 1152, 189, 189, 99, 594, 594, 198, 99, 198, 99, 594, 99, 378378, 45045, 99, 396, 30294, 2279277, 14256, 3267, 1089, 6336, 9702, 20196, 6534, 396, 11088, 20117097, 99, 99, 6318675, 594, 21978, 1089, 297, 2178, 594, 1683, 1485

Offset: 1

Labos Elemer, May 31 2002

			n=7, a(7)=GCD[5040,405]=45.

Cf. A055483.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[IntegerDigits[x]] Table[GCD[w!, tn[red[w! ]]], {w, 1, 128}]

a(n)=A055483[n! ]

A072033 Smallest x > 0 such that gcd(2^x, A004086(2^x)) = 2^n.

Original entry on oeis.org

4, 1, 2, 3, 26, 131, 227, 301, 567, 879, 3240, 11051, 8048, 38911, 7321, 97309, 108190, 6294, 138124, 4675268, 2687104, 1336154, 5774420

Offset: 1

Labos Elemer, Jun 07 2002

a(14)=7321, a(17)=6294.

			n=4: a(4)=26 because gcd(2^26, reverse(2^26)) = gcd(67108864, 46880176) = 16 = 2^n.

Cf. A055483, A004086, A071686, A072032.

a[n_] := Block[{k=1}, While[ IntegerExponent[ GCD[2^k, FromDigits@ Reverse@ IntegerDigits[2^k]], 2] != n, k++]; k]; Array[a, 13, 0] (* Giovanni Resta, Oct 28 2019 *)

a(n) = min{x: gcd(2^x, reverse(2^x))=2^n} = min{x: A055483(x)=2^n}.

A072032(a(n)) = 2^n.

Offset corrected, missing a(3) and a(13)-a(22) added by Giovanni Resta, Oct 28 2019

cp's OEIS Frontend

A072018 Numbers k for which gcd(k, reverse(k)) = 243 = 3^5.

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

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

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A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

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A071687 Non-palindromic numbers such that either x=q1.Rev[x] or Rev[x]=q2.x, where R[x]=A004086[x] and q1 or q2 are integers not divisible by 10.

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A069652 GCD of all the numbers obtained by permuting the digits of n.

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A072017 Numbers k such that gcd(k, reverse(k)) = 81 = 3^4, where reverse(x) = A004086(x).

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A337927 a(n) = n / GCD (n, reverse of n).

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A071678 GCD of n! and the reverse of n!.