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 A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1. 4
 1, 2, 2, 3, 3, 4, 8, 9 (list; graph; refs; listen; history; text; internal format)
 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 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 * A165120 A165129 A113773 Adjacent sequences:  A236207 A236208 A236209 * A236211 A236212 A236213 KEYWORD fini,full,nonn AUTHOR Jonathan Sondow, Jan 20 2014 STATUS approved

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Last modified August 18 02:34 EDT 2022. Contains 356204 sequences. (Running on oeis4.)