A134514 List of pairs (n,m) with n < m such that the decimal expansion of m is a cyclic shift of that of n and m^2 is a cyclic shift of n^2.

12, 21, 122, 221, 1222, 2221, 4615, 5461, 12222, 22221, 122222, 222221, 402046, 640204, 603069, 960306, 869041, 904186, 1222222, 2222221, 12222222, 22222221, 55887353, 58873535, 122222222, 222222221, 1222222222, 2222222221, 3672179309, 9367217930

Offset

1,1

Comments

Inspired by David W. Wilson's messages in Seqfan list.

Links

Table of n, a(n) for n=1..30.

Example

12 and 21 are rotationally connected and also their squares 144, 441 are obtained from each other by rotation of their decimal representations.
Also 122 and 221 are rotationally connected as well as their squares 14884 and 48841.
Notice infinite pattern (n,m)= (12...2, 2...21).
Corresponding squares:
{144, 441},
{14884, 48841},
{1493284, 4932841},
{21298225, 29822521},
{149377284, 493772841},
{14938217284, 49382172841},
{161640986116, 409861161616},
{363692218761, 922187613636},
{755232259681, 817552322596},
{1493826617284, 4938266172841}.
From Pieter Post, Jun 30, 2016: (Start)
There is another infinite subsequence:
The cyclic pair (201023, 320102)*k (for k = 2 and 3) and its squares (40410256529, 102465290404)*k^2.
The next in the sequence is:
(020001000203, 30200010002)*k,(400040009120406041209, 912040604120900040004)*k^2 (for k = 16, 17, ..., 33).
In general: (0{n}20{2n+1}10{2n+1}20{n}3, 30{n}20{2n+1}10{2n+1}20{n}3)* k, where lower bound of k = 5*10^(n-1)*sqrt(10) and upper bound of k = 3{n+1} for n = 0, 1, 2, 3, 4, etc.
For example, n = 6 gives lower bound k = 1581139 with lower cyclic pair: (316227800000001581139000000031622784743417, 474341731622780000000158113900000003162278)
and corresponding squares:(100000021492841000000214928422500007835889744785236492844223926514928486978524835889, 225000078358897447852364928442239265149284869785248358891000000214928410000002149284)
and upper bound k = 3333333 with upper cyclic pair:(666666600000003333333000000066666669999999, 999999966666660000000333333300000006666666)
and its corresponding squares:
(444444355555564444443555555699999993333332111111222222217777778888888766666660000001,
999999933333321111112222222177777788888887666666600000014444443555555644444435555556).
(End)

Crossrefs

Cf. A134584, A134585.

Sequence in context: A044005 A321771 A225864 * A030299 A352991 A268532

Adjacent sequences: A134511 A134512 A134513 * A134515 A134516 A134517

Keyword

base,nonn,tabf

Author

Zak Seidov, Jan 12 2008

Extensions

More terms from Max Alekseyev, Oct 14 2010

Status

approved