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A130179 Largest k such that k <= 81*(number of digits of k^n)*(number of digits of k^(n+1)). 5

%I

%S 2268,7776,18954,35397,56376,85050,119556,159894,209952,267300,331047,

%T 402084,479520,570807,670032,777195,892296,1015335,1146312,1285227,

%U 1432080,1586871,1749600,1932498,2125035,2312712,2522340,2741607

%N Largest k such that k <= 81*(number of digits of k^n)*(number of digits of k^(n+1)).

%C a(n) is an upper bound for A130181(n) and all the more so for A126783(n); apparently even A130181(n) < a(n)/4.

%C All terms are divisible by 81; the quotients a(n)/81 are in A130085.

%C For some n (18, 34, 35, 38, 42, 58, 59, ...) the line y = x and the graph of the staircase function y = 81*(number of digits of x^n)*(number of digits of x^(n+1)) intersect twice; this possibility has to be taken into account by the program.

%H Klaus Brockhaus, <a href="/A130179/b130179.txt">Table of n, a(n) for n=1..100</a>

%e Let D(n,k) = 81*(number of digits of k^n)*(number of digits of k^(n+1)).

%e D(2,k) > k for k = 1..4641, D(2,k) = 7776 for k = 4642..9999, D(2,k) < k for k >= 10000, hence a(2) = 7776.

%e D(18,k) > k for k = 1..885866, D(18,k) = 997272 for k = 885867..999999, D(18,k) = 1015335 for k = 1000000..1128837, D(18,k) < k for k >= 1128838, hence a(18) = 1015335.

%o (PARI) {for(n=1, 28, s=30*n; k=s; while(k<81*length(Str(k^n))*length(Str(k^(n+1))), k+=s); r=0; g=0; k-=s; b=1; while(b, p=81*length(Str(k^n))*length(Str(k^(n+1))); if(r<p, r=p; h=r; if(k>r, b=0, g=h)); k++); print1(g, ","))}

%Y Cf. A126783, A130181, A130085.

%K nonn,base

%O 1,1

%A _Klaus Brockhaus_, May 20 2007

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