A371886 The number of multiples (k) of n where an equal temperament with k*n divisions of the octave's best approximation of the just perfect fifth (3/2) is equal to that of an equal temperament with n divisions.

1, 2, 2, 1, 6, 1, 5, 1, 1, 3, 1, 25, 1, 2, 2, 1, 8, 1, 4, 1, 1, 3, 1, 12, 1, 2, 2, 1, 13, 1, 3, 1, 1, 4, 1, 8, 1, 2, 2, 1, 30, 1, 3, 1, 1, 5, 1, 6, 1, 2, 2, 1, 165, 1, 2, 2, 1, 6, 1, 5, 1, 1, 3, 1, 22, 1, 2, 2, 1, 9, 1, 4, 1, 1, 3, 1, 11, 1, 2, 2, 1, 15, 1, 3

Offset

1,2

Comments

"Best" is by minimum absolute difference, so the best approximation for n divisions is the rational r = f/n which minimizes d = abs(r - L) where L = log_2(3/2). The value of f for each n is A366701(n).

a(n) = k is the largest k for which A366701(k*n) = k * A366701(n); i.e., subdividing by k has not offered a better approximation than r.

This k is the largest k for which 1/(k*n) > 2*d, since a step to f +- 1/(k*n) is not in the range (L-d, L+d) and therefore is not closer to L.

Links

Soren Allen, Table of n, a(n) for n = 1..10000

Soren Allen, Plot of n, log(a(n)) for n = 1..3000; the asymptotic "peak-like" curves in this plot consist of values of n which differ by 53.

Soren Allen, Plot of n, log(a(n)) for n = 1..100000; the asymptotic "peak-like" curves in this plot consist of values of n which differ by 665.

Formula

a(n) = floor(1 / (2 * abs(round(n*log_2(3/2)) - n*log_2(3/2)))).

Example

For n = 12, the nearest integer to 12*log_2(3/2) is 7, yielding r = 7/12. Because r is equal for all multiples of 12 (24 yields 14/24, 36 yields 21/36, etc.) through 12*25=300 (175/300), a(12) = 25. 300 is in fact the highest multiple, as the best approximation for 12*26=312 is 183/312, which does not reduce to 7/12.

Mathematica

a[n_]:= Floor[1 / (2 * Abs[Round[n*Log2[3/2]] - n*Log2[3/2]])]; Array[a, 84] (* Stefano Spezia, Apr 12 2024 *)

Crossrefs

Cf. A366701.

Sequence in context: A371727 A326962 A350820 * A280245 A290013 A220963

Adjacent sequences: A371883 A371884 A371885 * A371887 A371888 A371889

Keyword

nonn

Author

Soren Allen, Apr 10 2024

Status

approved