A127857 -id:A127857 - cp's OEIS frontend

A127856 Positive integers n such that if n is a square, then so is r(n), where r(n) is the nonnegative integer defined by cyclic replacement of the digits d of n in base 12, that is, d->d+1 if d<11 and d->0 if d=11.

36, 4356, 617796, 910116, 2090916, 4588164, 39463524, 88849476, 173132964, 262245636, 266146596, 1404000900, 12792967236, 1491969417444, 1842170994756, 1904449680324, 7504323317604, 11853326293956, 265272427798596

Offset: 1

Walter Kehowski, Feb 04 2007

			a(1)=36 since 36=30 in base 12 and r(30)=41=7^2.
The only known element starting with the base-12 digit 11 (i.e., turning into a smaller square under the cyclic increments of the digits) is 549014564071245547886724. [From _Max Alekseyev_, Oct 24 2008]

Max Alekseyev, Table of n, a(n) for n=1..62

Cf. A117755, A127857, A127858, A127859, A127860, A127861.

A127858 Positive integers n such that r(n^2)=r(n)^2, where r is the cyclic replacement map of the digits d of n in base 12, that is, d->d+1 if d<11 and d->0 if d=11.

Original entry on oeis.org

6, 66, 786, 9426, 113106, 1357266, 16287186, 195446226, 2345354706, 28144256466, 337731077586, 4052772931026, 48633275172306, 583599302067666, 7003191624811986, 84038299497743826, 1008459593972925906

Offset: 1

Walter Kehowski, Feb 04 2007

In base 12 the sequence is 6, 56, 556, 5556, 55556, 555556, 5555556, 55555556, 555555556, 5555555556, and so on.

If r is the cyclic replacement map in base 10, then the only positive integers n with the property that r(n^2)=r(n)^2 appear to be 5, 45. For example, r(45^2)=r(2025)=3136=56^2=r(45)^2.

			a(2)=66 since, in base 12, 66=56, r(56)=67 and r(56^2)=r(2630)=3741=67^2.

Subsequence of A127857.

Cf. A117755, A127856, A127859, A127860, A127861, A192544.

a(n) = 5*(12^n - 1)/11 + 1. - Max Alekseyev, Jul 27 2023

Edited by Max Alekseyev, Jul 27 2023

A127859 a(n) = r(A127858(n)), where r if the cyclic replacement map of the digits d of n in base 12 defined by d->d+1 if d<11 and d->0 if d=11.

Original entry on oeis.org

7, 79, 943, 11311, 135727, 1628719, 19544623, 234535471, 2814425647, 33773107759, 405277293103, 4863327517231, 58359930206767, 700319162481199, 8403829949774383, 100845959397292591, 1210151512767511087

Offset: 1

Walter Kehowski, Feb 04 2007

In base 12 the sequence is 7, 67, 667, 6667, 66667, 666667, 6666667, 66666667, 666666667, 6666666667.

			a(2)=66 since, in base 12, 66=56, r(56)=67 and r(56^2)=r(2630)=3741=67^2.
In base 12, a(2)=r(A127858(2))=r(56)=67. In base 10, 67 is 79.

Cf. A117755, A127856, A127857, A127858, A127860, A127861, A192544.

a(n) = 6*(12^n - 1)/11 + 1. - Max Alekseyev, Jul 27 2023

More terms from Max Alekseyev, Jul 27 2023

A127860 a(n) = A127858(n)^2.

Original entry on oeis.org

36, 4356, 617796, 88849476, 12792967236, 1842170994756, 265272427798596, 38199227257643076, 5500688696956346436, 792099172023982809156, 114062280767400751587396, 16424968430457074953412676

Offset: 1

Walter Kehowski, Feb 04 2007

In base 12 the sequence is 30, 2630, 259630, 25909630, 2590409630, 259037409630, 259036X7409630, 259036X1X7409630, 259036X151X7409630, 259036X14851X7409630, where X is 10 and E is 11.

			a(1) = A127858(1)^2 = 6^2 = 36.

Cf. A117755, A127856, A127857, A127858, A127859, A127861, A192544.

a(n) = (5*(12^n - 1)/11 + 1)^2 = A127861(n) - (12^(2*n) - 1)/11.. - Max Alekseyev, Jul 27 2023

Edited by Max Alekseyev, Jul 27 2023

A127861 a(n) = A127859(n)^2.

Original entry on oeis.org

49, 6241, 889249, 127938721, 18421818529, 2652725580961, 381992288212129, 55006887157191841, 7920991722491368609, 1140622807701026002081, 164249684304894971368609, 23651954539856242601907361

Offset: 1

Walter Kehowski, Feb 04 2007

In base 12 the sequence is 41, 3741, 36X741, 36X1X741, 36X151X741, 36X14851X741, 36X147E851X741, 36X147E2E851X741, 36X147E262E851X741, 36X147E25962E851X741, where X is 10 and E is 11.

			a(1) = A127859(1)^2 = 7^2 = 49.

Cf. A117755, A127856, A127857, A127858, A127859, A127860, A192544.

a(n) = (6*(12^n - 1)/11 + 1)^2 = A127860(n) + (12^(2*n) - 1)/11. - Max Alekseyev, Jul 27 2023

Edited by Max Alekseyev, Jul 27 2023

A192544 Bases b such that all integers m having the commuting property r(m)^2 = r(m^2), where r is cyclic replacement of digits d->(d+1) mod b, are of the form m = (b/2 - 1)*(b^k - 1)/(b - 1) + 1 for k >= 1.

Original entry on oeis.org

8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264

Offset: 1

Walter Kehowski, Jul 04 2011

The bases b form the arithmetic sequence 8+4*k, k>=0, so b/2 is necessarily even. The bases b=2 and b=4 have b/2 as the only number with the commuting property. No odd base b has the commuting property.

			In base 8, the numbers with the commuting property are 4, 34, 334, 3334, 33334, 333334 etc, given by the formula 3*(8^k - 1)/7 + 1.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Except for initial terms, same as A008586 and A124354.

Cf. A059558, A192544, A117755, A127856, A127857, A127859, A127860, A127861.

a[n_] := 4*(n + 1); Table[a[n], {n, 1, 65}] (* Robert P. P. McKone, Aug 25 2023 *)

From Chai Wah Wu, Dec 29 2021: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 2.
G.f.: x*(8 - 4*x)/(x - 1)^2. (End)

More terms from Chai Wah Wu, Dec 29 2021

Edited by Max Alekseyev, Aug 24 2023

cp's OEIS Frontend

A127856 Positive integers n such that if n is a square, then so is r(n), where r(n) is the nonnegative integer defined by cyclic replacement of the digits d of n in base 12, that is, d->d+1 if d<11 and d->0 if d=11.

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A127858 Positive integers n such that r(n^2)=r(n)^2, where r is the cyclic replacement map of the digits d of n in base 12, that is, d->d+1 if d<11 and d->0 if d=11.

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A127859 a(n) = r(A127858(n)), where r if the cyclic replacement map of the digits d of n in base 12 defined by d->d+1 if d<11 and d->0 if d=11.

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A127860 a(n) = A127858(n)^2.

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A127861 a(n) = A127859(n)^2.

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A192544 Bases b such that all integers m having the commuting property r(m)^2 = r(m^2), where r is cyclic replacement of digits d->(d+1) mod b, are of the form m = (b/2 - 1)*(b^k - 1)/(b - 1) + 1 for k >= 1.

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