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A330733
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Triangle read by rows in which row n is the "complete rhythm" of n (see Comments for precise definition).
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0
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1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 2, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 0, 6, 0, 4, 2, 3, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,10
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COMMENTS
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Define the "natural rhythm" of any positive integer n to be the sequence consisting of n-1 zeros followed by 1; e.g., the natural rhythm of 5 is [0, 0, 0, 0, 1].
Define the "complete rhythm" of any positive integer n to be the term-by-term sum of the natural rhythm of n and the complete rhythm of f for every proper divisor f of n, extended through n/f cycles so as to give n terms. (Thus the complete rhythm of any noncomposite number is simply its natural rhythm.)
E.g., n=4 has a unique proper factor f=2 (whose complete rhythm is simply its natural rhythm, since 2 is prime).
Thus, for 4, we must add the following two components:
[0, 0, 0, 1] (the natural rhythm of 4)
+ [0, 1, 0, 1] (the rhythm of 2, repeated to give 4 terms)
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[0, 1, 0, 2] (the complete rhythm of 4).
Right diagonal is A002033 (conjectured).
Any prime column stripped of zeros also yields A002033 (conjectured).
Positions of 0 in each row n > 1 are in the reduced residue system of n (A038566). Therefore the number of zeros in each row n > 1 is given by the Euler totient function (A000010). This arises because a nonzero addend is introduced for multiples of divisors of n; the numbers k < n such that gcd(k,n) = 1 remain 0.
Conversely, nonzero positions in each row n > 1 are in the cototient of n (A121998), their number given by row n of A051953. (End)
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LINKS
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EXAMPLE
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Here are the rhythms of the first thirteen positive integers:
1 | 1
2 | 0, 1
3 | 0, 0, 1
4 | 0, 1, 0, 2
5 | 0, 0, 0, 0, 1
6 | 0, 1, 1, 1, 0, 3
7 | 0, 0, 0, 0, 0, 0, 1
8 | 0, 2, 0, 3, 0, 2, 0, 4
9 | 0, 0, 1, 0, 0, 1, 0, 0, 2
10 | 0, 1, 0, 1, 1, 1, 0, 1, 0, 3
11 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
12 | 0, 3, 2, 4, 0, 6, 0, 4, 2, 3, 0, 8
13 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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The complete rhythm of 12 is composed as follows:
12 has a "natural rhythm" of
12 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
12 has proper divisors 2, 3, 4 and 6, whose complete rhythms are
2 | 0, 1
3 | 0, 0, 1
4 | 0, 1, 0, 2
6 | 0, 1, 1, 1, 0, 3
When the padded (i.e., repeated) rhythms of the proper factors are added to the natural rhythm of 12, we have
2 | 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1
3 | 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1
4 | 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2
6 | 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 3
12 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
===+==============================================
12 | 0, 3, 2, 4, 0, 6, 0, 4, 2, 3, 0, 8
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MATHEMATICA
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Nest[Function[{a, n, d}, Append[#1, Total@ Map[PadRight[a[[#]], n, a[[#]] ] &, d] + Append[ConstantArray[0, n - 1], 1]]] @@ {#1, #2, Most@ Rest@ Divisors[#2]} & @@ {#, Length@ # + 1} &, {{1}}, 12] // Flatten (* Michael De Vlieger, Dec 29 2019 *)
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PROG
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(Python)
def memoize(f):
memo = {}
def helper(x):
if x not in memo:
memo[x] = f(x)
return memo[x]
return helper
@memoize
def unique_factors_of(n):
factors = []
for candidate in range(2, n//2 + 1):
if n % candidate == 0:
factors.append(candidate)
return factors
@memoize
def is_prime(n):
if n <= 1:
return False
if n <= 3:
return True
if n % 2 == 0 or n % 3 == 0:
return False
i = 5
while i * i <= n:
if n % i == 0 or n % (i + 2) == 0:
return False
i = i + 6
return True
@memoize
def rhythm(n):
if n == 0:
return [0]
natural_rhythm_of_n = [0]*(n-1)
natural_rhythm_of_n += [1]
if is_prime(n):
return natural_rhythm_of_n
else:
component_rhythms = [natural_rhythm_of_n]
for divisor in unique_factors_of(n):
component_rhythm = n//divisor * rhythm(divisor)
component_rhythms.append(component_rhythm)
return [sum(i) for i in zip(*component_rhythms)]
for i in range(0, 201):
formatted_string = f'{str(i).rjust(3)}|'
for note in rhythm(i):
formatted_string += f'{str(note).rjust(4)}'
print(formatted_string)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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