A134514 - OEIS

%I #24 Feb 10 2020 18:24:27

%S 12,21,122,221,1222,2221,4615,5461,12222,22221,122222,222221,402046,

%T 640204,603069,960306,869041,904186,1222222,2222221,12222222,22222221,

%U 55887353,58873535,122222222,222222221,1222222222,2222222221,3672179309,9367217930

%N 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.

%C Inspired by _David W. Wilson_'s messages in Seqfan list.

%e 12 and 21 are rotationally connected and also their squares 144, 441 are obtained from each other by rotation of their decimal representations.

%e Also 122 and 221 are rotationally connected as well as their squares 14884 and 48841.

%e Notice infinite pattern (n,m)= (12...2, 2...21).

%e Corresponding squares:

%e {144, 441},

%e {14884, 48841},

%e {1493284, 4932841},

%e {21298225, 29822521},

%e {149377284, 493772841},

%e {14938217284, 49382172841},

%e {161640986116, 409861161616},

%e {363692218761, 922187613636},

%e {755232259681, 817552322596},

%e {1493826617284, 4938266172841}.

%e From _Pieter Post_, Jun 30, 2016: (Start)

%e There is another infinite subsequence:

%e The cyclic pair (201023, 320102)*k (for k = 2 and 3) and its squares (40410256529, 102465290404)*k^2.

%e The next in the sequence is:

%e (020001000203, 30200010002)*k,(400040009120406041209, 912040604120900040004)*k^2 (for k = 16, 17, ..., 33).

%e 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.

%e For example, n = 6 gives lower bound k = 1581139 with lower cyclic pair: (316227800000001581139000000031622784743417, 474341731622780000000158113900000003162278)

%e and corresponding squares:(100000021492841000000214928422500007835889744785236492844223926514928486978524835889, 225000078358897447852364928442239265149284869785248358891000000214928410000002149284)

%e and upper bound k = 3333333 with upper cyclic pair:(666666600000003333333000000066666669999999, 999999966666660000000333333300000006666666)

%e and its corresponding squares:

%e (444444355555564444443555555699999993333332111111222222217777778888888766666660000001,

%e 999999933333321111112222222177777788888887666666600000014444443555555644444435555556).

%e (End)

%Y Cf. A134584, A134585.

%K base,nonn,tabf

%O 1,1

%A _Zak Seidov_, Jan 12 2008

%E More terms from _Max Alekseyev_, Oct 14 2010