OFFSET
1,2
COMMENTS
---------------------------Rhythmic structure example:---------------------------Enunciation: aababbccdbfe. Legend: a-a; b-aa; c-ba; d-bb; e-cc; f-db; g-fe. Unfolding: g (fe) (db)e db(cc) (bb)bcc bbb(ba)(ba) (aa)(aa)(aa)(aa)a(aa)a. "a" is a time unity. Each letter lasts the time of the sum of two others. The least common multiple of all letters is 12 (= a(6) in the sequence). This structure may represent alternated musical measures of 3/4 6/8.
Equivalently, this sequence corresponds to numbers that are the least common multiple of the terms of some addition chain. - Rémy Sigrist, May 23 2019
LINKS
Zizheng Fang, Table of n, a(n) for n = 1..1999
Zizheng Fang, Python program for generating A308115
EXAMPLE
1: divisors -- 1; binary sums -- 2; least common multiples -- 2.
2: divisors -- 1, 2; binary sums -- 2, 3, 4; least common multiples -- 4, 6.
4: divisors -- 1, 2, 4; binary sums -- 2, 3, 4, 5, 6, 8; least common multiples -- 6, 8, 12, 20, 24.
6: divisors -- 1, 2, 3, 6; binary sums -- 2, 3, 4, 5, 6, 7, 8, 9, 12; least common multiples -- 8, 12, 18, 20, 24, 30, 42.
8: divisors -- 1, 2, 4, 8; binary sums -- 2, 3, 4, 5, 6, 8, 9, 10, 12, 16; least common multiples -- 12, 16, 18, 20, 24, 30, 40, 42, 72.
PROG
Computing rhythmic numbers (generic draft program):
% let N be the variable set of proposed numbers
N = {1};
for n = 1 to (number of wanted terms)
% extract the least number
a(n) = (least number of N); N = N - {a(n)};
% make new proposed numbers
D = {divisors of a(n)};
D = D - {divisors that are not binary sum of others except 1}
for p = 1 to (cardinal of D); for q = 1 to p;
R = (least common multiple of {a(n), D(p) + D(q)});
if R > a(n) % a(n) must not be reinserted in N
% the element will be added to the set only if there isn't
N = N + {R};
next q; next p;
next n.
CROSSREFS
KEYWORD
nonn
AUTHOR
Carlos Palma Ramos, May 13 2019
EXTENSIONS
More terms from Rémy Sigrist, May 23 2019
STATUS
approved