A267122 Numbers n such that 1.5^n is closer to an integer than 1.5^m for any 0 < m < n.

1, 2, 4, 29, 46, 58, 95, 153, 157, 163, 455, 1060, 1256, 2677, 3328, 12429, 49304, 112896, 129638, 164000

Offset: 1

Author

Charles R Greathouse IV, Jan 11 2016

Keywords

• nonn

Comments

Zudilin proves that the distance from 1.5^n to the nearest integer is at least 0.5803^n for large enough n; it seems that n > 4 suffices. (The "large enough" constant in the proof is effective but not explicit.)

Examples

			1.5^29 = 127834.039... which is within 0.039... of an integer, yielding a(4) = 29.
1.5^46 = 125949925.968... which is within 0.031... of an integer, yielding a(5) = 46.
		

Links

• Wadim Zudilin, A new lower bound for ∥(3/2)^k∥, J. Théor. Nombres Bordeaux 19:1 (2007), pp. 311-323.

Crossrefs

Cf. A079490, A080052.

Programs

PARI

f(x)=x=frac(x); if(x>1/2,1-x,x)
t=r=1;for(n=1,1e6, tt=f(t*=3/2); if(tt