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A308115
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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.
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2
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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
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OFFSET
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1,2
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COMMENTS
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---------------------------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
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LINKS
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EXAMPLE
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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.
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PROG
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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.
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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