A308115 - OEIS

%I #29 Jan 30 2020 04:22:10

%S 1,2,4,6,8,12,16,18,20,24,30,32,36,40,42,48,54,60,64,72,80,84,90,96,

%T 100,108,120,126,128,140,144,150,156,160,162,168,180,192,198,200,210,

%U 216,220,240,252,256,264,270,272,280,288,294,300,312,320,324,330,336

%N Rhythmic numbers: each number in this list is the least common multiple of some set of numbers S such that 1 is in S and every element of S except for 1 is a sum of two other elements of S.

%C ---------------------------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.

%C Equivalently, this sequence corresponds to numbers that are the least common multiple of the terms of some addition chain. - _Rémy Sigrist_, May 23 2019

%H Zizheng Fang, <a href="/A308115/b308115.txt">Table of n, a(n) for n = 1..1999</a>

%H Zizheng Fang, <a href="/A308115/a308115.txt">Python program for generating A308115</a>

%e 1: divisors -- 1; binary sums -- 2; least common multiples -- 2.

%e 2: divisors -- 1, 2; binary sums -- 2, 3, 4; least common multiples -- 4, 6.

%e 4: divisors -- 1, 2, 4; binary sums -- 2, 3, 4, 5, 6, 8; least common multiples -- 6, 8, 12, 20, 24.

%e 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.

%e 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.

%o Computing rhythmic numbers (generic draft program):

%o % let N be the variable set of proposed numbers

%o N = {1};

%o for n = 1 to (number of wanted terms)

%o % extract the least number

%o a(n) = (least number of N); N = N - {a(n)};

%o % make new proposed numbers

%o D = {divisors of a(n)};

%o D = D - {divisors that are not binary sum of others except 1}

%o for p = 1 to (cardinal of D); for q = 1 to p;

%o   R = (least common multiple of {a(n), D(p) + D(q)});

%o   if R > a(n) % a(n) must not be reinserted in N

%o    % the element will be added to the set only if there isn't

%o    N = N + {R};

%o next q; next p;

%o next n.

%Y The present sequence is a subset of A174973 and the first 22 terms are equal. With divisors of 88 (= A174973(23)) we can't make any of 11, 22, 44, 88 adding two of 1, 2, 4, 8.

%K nonn

%O 1,2

%A _Carlos Palma Ramos_, May 13 2019

%E More terms from _Rémy Sigrist_, May 23 2019