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 A330733 Triangle read by rows in which row n is the "complete rhythm" of n (see Comments for precise definition). 0
 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)
 OFFSET 1,10 COMMENTS 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) ==============   [0, 1, 0, 2]  (the complete rhythm of 4). Right diagonal is A002033 (conjectured). Any prime column stripped of zeros also yields A002033 (conjectured). From Michael De Vlieger, Dec 29 2019: (Start) 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) LINKS EXAMPLE 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 . 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 MATHEMATICA 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 *) PROG (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) CROSSREFS Cf. A002033 (number of perfect partitions of n), A000040 (prime numbers), A000010, A038566, A051953, A121998. Sequence in context: A101668 A141846 A188171 * A328496 A035202 A227835 Adjacent sequences:  A330730 A330731 A330732 * A330734 A330735 A330736 KEYWORD nonn,tabl AUTHOR Andrew Hood, Dec 28 2019 EXTENSIONS Name clarified by Omar E. Pol and Jon E. Schoenfield, Dec 31 2019 STATUS approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)