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%I #27 Sep 18 2012 20:08:52
%S 126,133,63,32,26,27,42,33,1,16,15,11,76,15,26,19,18,8,1,45,38,19,12,
%T 16,30,22,11,21,1,16,16,11,12,11,13,10,23,10,1,22,19,6,18,25,23,11,10,
%U 6,1,6,8,20,14,17,11,13,14,13,1,15,14,17,21,16,16,9,4,11
%N a(n) is the conjectured highest power of n which has no two identical digits in succession.
%C Contribution from _Charles R Greathouse IV_, Sep 17 2012: (Start)
%C a(n) = 0 for infinitely many n; such n have positive density in this sequence. Question: are such n of density 1?
%C A naive heuristic suggests that there are infinitely many n such that a(n) = 6 but only finitely many a(n) such that a(n) > 6. This suggests a weaker conjecture: this sequence is bounded. (end)
%H V. Raman and T. D. Noe, <a href="/A216063/b216063.txt">Table of n, a(n) for n = 2..1000</a> (V. Raman computed the terms 2 to 99)
%e 3^133 = 2865014852390475710679572105323242035759805416923029389510561523 which has no two adjacent identical digits.
%t Table[mx = 0; Do[If[! MemberQ[Differences[IntegerDigits[n^k]], 0], mx = k], {k, 1000}]; mx, {n, 2, 100}] (* _T. D. Noe_, Sep 17 2012 *)
%o (PARI) isA043096(n)=my(v=digits(n));for(i=2,#v,if(v[i]==v[i-1],return(0)));1
%o a(n)=my(best=0); if(n==14,76,for(k=1, max(9,94\sqrt(log(n))), if(isA043096(n^k), best=k)); best ) \\ (conjectural) _Charles R Greathouse IV_, Sep 17 2012
%Y Cf. A043096, A216064, A216065, A215236.
%K nonn,base
%O 2,1
%A _V. Raman_, Sep 01 2012
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