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

Author

Carlos Palma Ramos, May 13 2019

Keywords

• nonn

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

Examples

			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.
		

Links

• Zizheng Fang, Table of n, a(n) for n = 1..1999
• Zizheng Fang, Python program for generating A308115

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.

Extensions

More terms from Rémy Sigrist, May 23 2019