A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.

1, 2, 2, 3, 3, 4, 8, 9

Offset

1,2

Comments

Philippe de Vitry (1291-1361), a musician from Vitry-en-Artois 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 (1288-1344) 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), 231-234.

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 * A357416 A362251 A165120

Adjacent sequences: A236207 A236208 A236209 * A236211 A236212 A236213

Keyword

fini,full,nonn

Author

Jonathan Sondow, Jan 20 2014

Status

approved