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

2268, 7776, 18954, 35397, 56376, 85050, 119556, 159894, 209952, 267300, 331047, 402084, 479520, 570807, 670032, 777195, 892296, 1015335, 1146312, 1285227, 1432080, 1586871, 1749600, 1932498, 2125035, 2312712, 2522340, 2741607

Offset: 1

Author

Klaus Brockhaus, May 20 2007

Keywords

• nonn
• base

Comments

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

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

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.

Examples

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

Links

• Klaus Brockhaus, Table of n, a(n) for n=1..100

Crossrefs

Cf. A126783, A130181, A130085.

Programs

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(rr, b=0, g=h)); k++); print1(g, ","))}