A081464 Numbers k such that the fractional part of (3/2)^k decreases monotonically to zero.

1, 2, 4, 29, 95, 153, 532, 613, 840, 2033, 2071, 3328, 12429, 112896, 129638, 371162, 1095666, 3890691, 4264691, 31685458, 61365215, 92432200, 144941960

Offset: 1

Author

Benoit Cloitre, Apr 21 2003

Keywords

• nonn
• more

Comments

Do the values characterize 3/2? If not, what set do they characterize? - Bill Gosper, Jul 03 2008

Comments from Eliora Ben-Gurion, Dec 26 2021: (Start)

The numbers have an interpretation in terms of music theory - these numbers characterize integer harmonics that offer monotonically closer approximations to the stacks of just-intonated perfect fifths (3/2). Repeated stacking of this interval forms the basis of the Pythagorean tuning. For example, a(3) = 4; 1.5^4 = 5.0625, therefore the 5th harmonic is close to a stack of 4 perfect fifths. This specific difference is known as the syntonic comma.

Likewise, 1.5^29 = 127834.039..., therefore the 127834th harmonic is close to a stack of 29 perfect fifths, but in real life this example is wider than the human hearing range (20 Hz to 20 kHz, 1000 times), therefore lacks practical application. (End)

Programs

Mathematica

a = 1; Do[b = N[ Mod[(3/2)^n, 1]]; If[b < a, Print[n]; a = b], {n, 1, 10^6}]

PARI

x=1; y=1; a(n)=if(n<0,0,b=y+1; while(frac((3/2)^b)>frac((3/2)^x),b++); x=b; y=b; b)

Extensions

More terms from Robert G. Wilson v, Apr 22 2003

a(16)-a(23) from Robert Gerbicz, Nov 21 2010