A227140 a(n) = n/gcd(n,2^5), n >= 0.

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 2, 65, 33, 67, 17, 69, 35

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

Cf. A000265, A227042, A227107, A106617, A276234.

Sequence in context: A161955 A276234 A000265 * A106617 A040026 A106609

Adjacent sequences: A227137 A227138 A227139 * A227141 A227142 A227143

Keyword

nonn,frac,easy,mult

Author

Wolfdieter Lang, Jul 04 2013

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

Status

approved