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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.

%I A192544 #25 Aug 19 2025 03:29:41
%S A192544 8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,
%T A192544 100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,160,164,
%U A192544 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
%N 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.
%C A192544 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.
%H A192544 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A192544 From _Chai Wah Wu_, Dec 29 2021: (Start)
%F A192544 a(n) = 2*a(n-1) - a(n-2) for n > 2.
%F A192544 G.f.: x*(8 - 4*x)/(x - 1)^2. (End)
%e A192544 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.
%t A192544 a[n_] := 4*(n + 1); Table[a[n], {n, 1, 65}] (* _Robert P. P. McKone_, Aug 25 2023 *)
%Y A192544 Except for initial terms, same as A008586 and A124354.
%Y A192544 Cf. A059558, A192544, A117755, A127856, A127857, A127859, A127860, A127861.
%K A192544 nonn,base,easy
%O A192544 1,1
%A A192544 _Walter Kehowski_, Jul 04 2011
%E A192544 More terms from _Chai Wah Wu_, Dec 29 2021
%E A192544 Edited by _Max Alekseyev_, Aug 24 2023