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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

Table of n, a(n) for n=1..91.

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.)