A308115 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.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 126, 128, 140, 144, 150, 156, 160, 162, 168, 180, 192, 198, 200, 210, 216, 220, 240, 252, 256, 264, 270, 272, 280, 288, 294, 300, 312, 320, 324, 330, 336

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

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.

Sequence in context: A325781 A231565 A191743 * A331828 A377308 A258118

Adjacent sequences: A308112 A308113 A308114 * A308116 A308117 A308118

Keyword

nonn

Author

Carlos Palma Ramos, May 13 2019

Extensions

More terms from Rémy Sigrist, May 23 2019

Status

approved