

A236210


Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.


4




OFFSET

1,2


COMMENTS

Philippe de Vitry (12911361), a musician from VitryenArtois in France, called numbers of the form 2^m * 3^n "harmonic numbers". He asked if all powers of 2 and 3 differ by more than 1 except the pairs 1 and 2, 2 and 3, 3 and 4, 8 and 9 (which correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone). Levi Ben Gerson (12881344) answered yes by proving that 3^n + 1 is not a power of 2 if n > 2; see A235365, A235366.


REFERENCES

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.


LINKS

Table of n, a(n) for n=1..8.
A. Herschfeld, The equation 2^x  3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231234.
H. Lenstra, Harmonic Numbers, MSRI, 1998.
J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
I. Peterson, Medieval Harmony, Math Trek, MAA, 2012.
Wikipedia, Gersonides
Wikipedia, Philippe de Vitry


EXAMPLE

8 + 1 = 2^3 + 1 = 3^2 = 9, so the pair 8 and 9 is in the sequence.


CROSSREFS

Cf. A003586, A006899, A061987, A108906, A235365, A235366.
Sequence in context: A343659 A146922 A283363 * A165120 A165129 A113773
Adjacent sequences: A236207 A236208 A236209 * A236211 A236212 A236213


KEYWORD

fini,full,nonn


AUTHOR

Jonathan Sondow, Jan 20 2014


STATUS

approved



