A071686 -id:A071686 - cp's OEIS frontend

A072005 Smallest solution to gcd(k, reverse(k)) = 3^n.

1, 3, 9, 2889, 2899999989, 4899999987, 19899999972, 29898999693, 49989958299, 49999917897, 99884394999, 372797889885, 1989767716659, 2678052898989, 17902896898419, 137530987695297, 189281899170567, 368055404997498, 14048104419899757, 437893473401621955, 218264275944702783

Offset: 0

Labos Elemer, Jun 04 2002

			n=4: 3^4 = 81, a(4) = 2899999989 = 3*3*3*3*35802469, reverse(a(4)) = 2*3*3*3*3*61111111; gcd = 81 = 3^n.

Sean A. Irvine, Table of n, a(n) for n = 0..25

Cf. A004086, A055483, A069554, A071686, A071686, A072021, A072050.

a(n) = A069554(3^n).

a(15)-a(20) from Giovanni Resta, Oct 30 2019

A072021 Smallest solution to gcd(x, reverse(x)) = 5^n.

Original entry on oeis.org

5, 5200, 521000, 5213750, 521875, 5218750, 52130234375, 5734841796875, 57869714843750, 526046650390625, 5265674365234375, 52187008544921875, 526515306396484375, 5213023309008789062500, 5213596736358642578125, 5260466086273193359375, 526041911745452880859375

Offset: 1

Labos Elemer, Jun 06 2002

			For n = 4, gcd(521875, 578125) = 3125 = 5^4.
For n = 8, a(8) = 5734841796875 = 5^9*2936239, reverse(a(8)) = 5786971484375 = 5^8*71*208657.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..25

Cf. A004086, A055483, A069554, A071686, A072005, A072050, A072016-A072018.

a(n) = {my(k = 1); while (gcd(k, fromdigits(Vecrev(digits(k)))) != 5^n, k++); k;} \\ Michel Marcus, Jul 13 2018

a(n) = A069554(5^n).

a(9)-a(18) from Hiroaki Yamanouchi, Sep 10 2014

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

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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.

A072032 a(n) = gcd(2^n, reverse(2^n)) = gcd(2^n, A004086(2^n)) = A055483(2^n).

Original entry on oeis.org

2, 4, 8, 1, 1, 2, 1, 4, 1, 1, 2, 8, 2, 1, 1, 4, 1, 2, 1, 1, 2, 2, 2, 1, 1, 16, 1, 2, 1, 1, 4, 4, 2, 1, 1, 2, 1, 8, 1, 1, 8, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 8, 4, 1, 1, 1, 1, 1, 8, 1, 1, 16, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 8, 1, 1, 1, 1, 1, 1, 8, 1, 2, 2, 1, 1, 1, 1, 1, 1, 16, 1, 8, 1, 1

Offset: 1

Labos Elemer, Jun 07 2002

			n=12: a(12) = gcd(4096,6904) = 8.

Indranil Ghosh, Table of n, a(n) for n = 1..31180

Cf. A004086, A055483, A071686.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] red[x_] := Reverse[IntegerDigits[x]] Table[GCD[2^w, tn[red[2^w]]], {w, 1, 128}]

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.

A071788 Integer quotients arising in A071687.

Original entry on oeis.org

34, 12, 45, 9, 175, 4, 175, 208, 34, 76, 12, 175, 175, 4, 45, 9, 9, 4, 175, 375, 175, 34, 232, 64, 12, 34, 208, 12, 15, 175, 45, 4, 45, 9, 9, 175, 25, 4, 175, 175, 175, 175, 375, 175, 175, 175, 175, 175, 34, 172, 4168, 208, 12, 34, 1528, 232, 76, 208, 12, 38125

Offset: 1

Labos Elemer, Jun 03 2002

			x=4200, Rev[x]=24, q=175.

Cf. A004086, A071686, A071687.

Quotients of q=Max[x/Rev[x], Rev[x], x], q>1 and Mod[q, 10]>0, I.e. integer quotient is neither 1, nor divisible by base=10. Rev[x]=A004086[x], digit reversal of x.

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

A072051 Smallest k such that gcd(k, reverse(k)) = 11^n.

Original entry on oeis.org

11, 121, 1331, 14641, 121110352, 1332213872, 105923336431682, 4676049710123077, 36606937477221265, 30983951005022964839, 1365869521861436622239

Offset: 1

Labos Elemer, Jun 10 2002

			a(8) = 4676049710123077 = (11^8)*13*1678009, reverse(a(8)) = 7703210179406764 = (11^8)*2*2*157*57223.

Cf. A004086, A055483, A069554, A071686 (=2^n), A072005 (=3^n), A072021 (=5^n), A072050 (=7^n).

a(n) = A069554(11^n).

a(9)-a(11) from Sean A. Irvine, Sep 02 2024

cp's OEIS Frontend

A072005 Smallest solution to gcd(k, reverse(k)) = 3^n.

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A072021 Smallest solution to gcd(x, reverse(x)) = 5^n.

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

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

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

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A072032 a(n) = gcd(2^n, reverse(2^n)) = gcd(2^n, A004086(2^n)) = A055483(2^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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A071788 Integer quotients arising in A071687.

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

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