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

Author

Wolfdieter Lang, Jul 04 2013

Keywords

• nonn
• frac
• easy
• mult

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).

Crossrefs

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

Programs

GAP

List([0..80], n-> n/Gcd(n, 2^5)); # G. C. Greubel, Feb 27 2019

Magma

[n/GCD(n, 2^5): n in [0..80]]; // G. C. Greubel, Feb 27 2019

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 *)

PARI

a(n)=n/gcd(n, 2^5); \\ Andrew Howroyd, Jul 23 2018

Sage

[n/gcd(n,2^5) for n in (0..80)] # G. C. Greubel, Feb 27 2019

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)

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018