A109053 -id:A109053 - cp's OEIS frontend

A051724 Numerator of n/12.

0, 1, 1, 1, 1, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 4, 17, 3, 19, 5, 7, 11, 23, 2, 25, 13, 9, 7, 29, 5, 31, 8, 11, 17, 35, 3, 37, 19, 13, 10, 41, 7, 43, 11, 15, 23, 47, 4, 49, 25, 17, 13, 53, 9, 55, 14, 19, 29, 59, 5, 61, 31, 21, 16, 65, 11, 67, 17, 23, 35, 71

Offset: 0

N. J. A. Sloane

Or, numerator of n/(n+12).
Arises in the equal-tempered musical scale - see Goldstein (1977), Table 1. - N. J. A. Sloane, Aug 29 2018
A strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 269.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 23.
A. A. Goldstein, Optimal temperament, SIAM Review 19.3 (1977): 554-562.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A000010, A109053.

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/12)); # Muniru A Asiru, Feb 24 2019

[Numerator(n/12): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/12),n=0..100); # Nathaniel Johnston, Apr 18 2011

f[n_]:=Numerator[n/(n+12)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)

a(n) = numerator(n/12); \\ Michel Marcus, Aug 19 2018

[lcm(n,12)/12 for n in range(0, 72)] # Zerinvary Lajos, Jun 09 2009

From David W. Wilson, Jun 12 2005: (Start)
a(n) = n/gcd(n, 12).
Multiplicative with a(2^e) = 2^max(0, e-2), a(3^e) = 3^max(0,e-1), a(p^e) = p^e otherwise. (End)
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109053(n)/12.
Dirichlet g.f.: zeta(s-1)*(1 - 2/3^s - 1/2^s + 2/6^s - 1/4^s + 2/12^s). (End)
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,12) is a quasi-polynomial in n since gcd(n,12) is a purely periodic sequence of period 12.
O.g.f.: F(x) - F(x^2) - 2*F(x^3) - F(x^4) + 2*F(x^6) + 2*F(x^12), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 12} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/3)*log(1/(1 - x^3)) + (2/4)*log(1/(1 - x^4)) + (2/6)*log(1/(1 - x^6)) + (4/12)*log(1/(1 - x^12)), where phi(n) denotes the Euler totient function A000010. (End)
Sum_{k=1..n} a(k) ~ (77/288) * n^2. - Amiram Eldar, Nov 25 2022

A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 4, 3, 4, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 6, 15, 4, 15, 6, 7, 0, 0, 8, 14, 6, 20, 20, 6, 14, 8, 0, 0, 9, 8, 21, 12, 5, 12, 21, 8, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 0

Offset: 0

Mitch Harris, Jun 18 2005

			Seen as an array:
  [0] 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  [1] 0, 1,  2,  3,  4,  5,  6,  7,  8,  9, ...
  [2] 0, 2,  2,  6,  4, 10,  6, 14,  8, 18, ...
  [3] 0, 3,  6,  3, 12, 15,  6, 21, 24,  9, ...
  [4] 0, 4,  4, 12,  4, 20, 12, 28,  8, 36, ...
  [5] 0, 5, 10, 15, 20,  5, 30, 35, 40, 45, ...
  [6] 0, 6,  6,  6, 12, 30,  6, 42, 24, 18, ...
  [7] 0, 7, 14, 21, 28, 35, 42,  7, 56, 63, ...
  [8] 0, 8,  8, 24,  8, 40, 24, 56,  8, 72, ...
  [9] 0, 9, 18,  9, 36, 45, 18, 63, 72,  9, ...
.
Seen as a triangle T(n, k) = lcm(n - k, k).
  [0] 0;
  [1] 0, 0;
  [2] 0, 1,  0;
  [3] 0, 2,  2,  0;
  [4] 0, 3,  2,  3,  0;
  [5] 0, 4,  6,  6,  4,  0;
  [6] 0, 5,  4,  3,  4,  5, 0;
  [7] 0, 6, 10, 12, 12, 10, 6,  0;
  [8] 0, 7,  6, 15,  4, 15, 6,  7, 0;
  [9] 0, 8, 14,  6, 20, 20, 6, 14, 8, 0;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows A000027, A109043, A109044, A109045, A109046, A109047, A109048, A109049, A109050, A109051, A109052, A109053, A006580 (row sums of triangle), A001477 (main diagonal, central terms).

Variants: A003990 is (1, 1) based, A051173 (T(n,k) = lcm(n,k)).

T := (n, k) -> ilcm(n - k, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 24 2025

lcm(n, k) = n*k / gcd(n, k) for (n, k) != (0, 0).

A070292 a(n) = lcm(12,n)/gcd(12,n).

Original entry on oeis.org

12, 6, 4, 3, 60, 2, 84, 6, 12, 30, 132, 1, 156, 42, 20, 12, 204, 6, 228, 15, 28, 66, 276, 2, 300, 78, 36, 21, 348, 10, 372, 24, 44, 102, 420, 3, 444, 114, 52, 30, 492, 14, 516, 33, 60, 138, 564, 4, 588, 150, 68, 39, 636, 18, 660, 42, 76, 174, 708, 5, 732, 186, 84, 48

Offset: 1

Amarnath Murthy, May 10 2002

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A070290, A070291, A070293, A109015, A109053.

for(n=1,100,print1(lcm(12,n)/gcd(n,12),","))

Vec(x*(12 + 6*x + 4*x^2 + 3*x^3 + 60*x^4 + 2*x^5 + 84*x^6 + 6*x^7 + 12*x^8 + 30*x^9 + 132*x^10 + x^11 + 132*x^12 + 30*x^13 + 12*x^14 + 6*x^15 + 84*x^16 + 2*x^17 + 60*x^18 + 3*x^19 + 4*x^20 + 6*x^21 + 12*x^22) / (x^24 - 2*x^12 + 1) + O(x^60)) \\ Colin Barker, Mar 05 2019

from math import gcd, lcm
def a(n): return lcm(12, n)//gcd(12, n)
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Dec 06 2021

a(n) = A109053(n)/A109015(n) = 12*n/A109015(n)^2. - R. J. Mathar, Feb 12 2019
a(n) = 2*a(n-12) - a(n-24). - R. J. Mathar, Feb 12 2019
G.f.: x*(12 + 6*x + 4*x^2 + 3*x^3 + 60*x^4 + 2*x^5 + 84*x^6 + 6*x^7 + 12*x^8 + 30*x^9 + 132*x^10 + x^11 + 132*x^12 + 30*x^13 + 12*x^14 + 6*x^15 + 84*x^16 + 2*x^17 + 60*x^18 + 3*x^19 + 4*x^20 + 6*x^21 + 12*x^22) / (x^24 - 2*x^12 + 1). - Colin Barker, Mar 05 2019
Sum_{k=1..n} a(k) ~ (703/288)*n^2. - Amiram Eldar, Oct 07 2023

More terms from Benoit Cloitre, May 16 2002

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A051724 Numerator of n/12.

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A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

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A070292 a(n) = lcm(12,n)/gcd(12,n).

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