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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac
• mult

Comments

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

References

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

Links

• 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).

Crossrefs

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).

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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