

A051724


Numerator of n/12.


25



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
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OFFSET

0,6


COMMENTS

Or, numerator of n/(n+12).
Multiplicative with a(2^e) = 2^max(0, e2), a(3^e) = 3^max(0,e1), a(p^e) = p^e otherwise.  David W. Wilson, Jun 12 2005
Arises in the equaltempered 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. AddisonWesley, Reading, MA, 1990, p. 269.


LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
P. Bala, A note on the sequence of numerators of a rational function
A. A. Goldstein, Optimal temperament, SIAM Review 19.3 (1977): 554562.


FORMULA

a(n) = n/gcd(n, 12).  David W. Wilson, Jun 12 2005
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109053(n)/12.
Dirichlet g.f.: zeta(s1)*(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 quasipolynomial 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)


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*)


PROG

(Sage) [lcm(n, 12)/12 for n in xrange(0, 72)] # Zerinvary Lajos, Jun 09 2009
(MAGMA) [Numerator(n/12): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011
(PARI) a(n) = numerator(n/12); \\ Michel Marcus, Aug 19 2018
(GAP) List([0..80], n>NumeratorRat(n/12)); # Muniru A Asiru, Feb 24 2019


CROSSREFS

Cf. 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).
Sequence in context: A319684 A102778 A135544 * A084303 A011508 A195429
Adjacent sequences: A051721 A051722 A051723 * A051725 A051726 A051727


KEYWORD

nonn,easy,frac,mult


AUTHOR

N. J. A. Sloane


STATUS

approved



