OFFSET
0,4
COMMENTS
H(n,4) = 2*n*4/(n+4) is the harmonic mean of n and 4. For n >= 4 the denominator of H(n,4) is (n+4)/gcd(8*n,n+4) = (n+4)/gcd(n+4,32). a(n+8) = A227042(n+4,4), n >= 0. The numerator of H(n,4) is given in A227107. Thus a(n) is related to denominator of the harmonic mean H(n-4, 4).
Note the difference from A000265(n) (odd part of n) = n/gcd(n,2^n), n >= 1, which differs for the first time for n = 64. a(64) = 2, not 1.
A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
FORMULA
a(n) = n/gcd(n, 2^5).
a(n) = denominator(8*(n-4)/n), n >= 0 (with denominator(infinity) = 0).
From Peter Bala, Feb 27 2019: (Start)
a(n) = numerator(n/(n + 32)).
O.g.f.: F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16) - F(x^32), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
From Bernard Schott, Mar 02 2019: (Start)
a(n) = 1 iff n is 1, 2, 4, 8, 16, 32 and a(2^n) = 2^(n-5) for n >= 5.
a(n) = n iff n is odd (A005408). (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^(e-min(e,5)), and a(p^e) = p^e for p > 2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s - 1/2^(2*s) - 1/2^(3*s) - 1/2^(4*s) - 1/2^(5*s)).
Sum_{k=1..n} a(k) ~ (683/2048) * n^2. (End)
MAPLE
seq(n/igcd(n, 32), n=0..70); # Muniru A Asiru, Feb 28 2019
MATHEMATICA
With[{c=2^5}, Table[n/GCD[n, c], {n, 0, 70}]] (* Harvey P. Dale, Feb 16 2018 *)
PROG
(PARI) a(n)=n/gcd(n, 2^5); \\ Andrew Howroyd, Jul 23 2018
(Magma) [n/GCD(n, 2^5): n in [0..80]]; // G. C. Greubel, Feb 27 2019
(Sage) [n/gcd(n, 2^5) for n in (0..80)] # G. C. Greubel, Feb 27 2019
(GAP) List([0..80], n-> n/Gcd(n, 2^5)); # G. C. Greubel, Feb 27 2019
CROSSREFS
KEYWORD
nonn,frac,easy,mult
AUTHOR
Wolfdieter Lang, Jul 04 2013
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Jul 23 2018
STATUS
approved