A153694 Numbers k such that the fractional part of (10/9)^k is less than 1/k.

1, 2, 7, 62, 324, 1647, 3566, 5464, 8655, 8817, 123956, 132891, 182098, 566593, 2189647, 2189648, 3501843, 3501844

Offset: 1

Author

Hieronymus Fischer, Jan 06 2009

Keywords

• nonn
• more

Comments

Numbers k such that fract((10/9)^k) < 1/k, where fract(x) = x-floor(x).

The next such number must be greater than 2*10^5.

a(19) > 10^7. - Robert Price, Mar 24 2019

Given a number k that is not only a term of this sequence but also has the property that the integer part of (10/9)^k is divisible by 9, we can expect that k+1 will likely also be a term of the sequence. E.g., k = 2189647 is a term because fract((10/9)^k) = 0.000000373557... < 0.000000456694... = 1/k, and since floor((10/9)^k) is divisible by 9, the integer and fractional parts of (10/9)^(k+1) will be exactly 10/9 times the integer and fractional parts of (10/9)^k, respectively, yielding a fractional part (10/9) * 0.000000373557... = 0.000000415064... < 0.000000456694... = 1/(k+1), so k+1 = 2189648 is also a term. - Jon E. Schoenfield, Mar 24 2019

Examples

			a(3) = 7 since fract((10/9)^7) = 0.09075... < 1/7, but fract((10/9)^k) >= 1/k for 3 <= k <= 6.
		

Crossrefs

Cf. A153662, A153670, A153678, A153686, A153698, A154130, A153702, A153710, A153718.

Programs

Mathematica

Select[Range[1000], FractionalPart[(10/9)^#] < (1/#) &] (* G. C. Greubel, Aug 24 2016 *)

Extensions

a(14)-a(18) from Robert Price, Mar 24 2019