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A081464
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Numbers k such that the fractional part of (3/2)^k decreases monotonically to zero.
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22
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1, 2, 4, 29, 95, 153, 532, 613, 840, 2033, 2071, 3328, 12429, 112896, 129638, 371162, 1095666, 3890691, 4264691, 31685458, 61365215, 92432200, 144941960
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OFFSET
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1,2
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COMMENTS
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Do the values characterize 3/2? If not, what set do they characterize? - Bill Gosper, Jul 03 2008
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)
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LINKS
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MATHEMATICA
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a = 1; Do[b = N[ Mod[(3/2)^n, 1]]; If[b < a, Print[n]; a = b], {n, 1, 10^6}]
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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