A276234 a(n) = n/gcd(n, 256).

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, 1, 65, 33, 67, 17, 69, 35, 71

Offset: 1

Author

Artur Jasinski, Aug 24 2016

Keywords

• nonn
• easy
• mult

Comments

a(n) first differs from A000265(n) at n = 512. - Andrew Howroyd, Jul 23 2018

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 = 1..1000
• Peter Bala, A note on the sequence of numerators of a rational function, 2019.
• Index entries for linear recurrences with constant coefficients, order 512.

Crossrefs

Cf. A276233 (numerators), A227140, A000265, A106617.

Programs

GAP

List([1..80], n-> n/Gcd(n, 2^8)); # G. C. Greubel, Feb 27 2019

Magma

[n/GCD(n, 2^8): n in [1..80]]; // G. C. Greubel, Feb 27 2019

Maple

seq(n/igcd(n,256), n=1..100); # Robert Israel, Aug 26 2016

Mathematica

Table[n/GCD[n, 2^8], {n,1,80}] (* G. C. Greubel, Feb 27 2019 *)

PARI

a(n)=n/gcd(n,256) \\ Charles R Greathouse IV, Aug 26 2016

Sage

[n/gcd(n, 2^8) for n in (1..80)] # G. C. Greubel, Feb 27 2019

Formula

a(2k-1) = 2k-1.

G.f.: (x+x^3)/(1-x^2)^2 +(x^2+x^6)/(1-x^4)^2 +(x^4+x^12)/(1-x^8)^2 +(x^8+x^24)/(1-x^16)^2 +(x^16+x^48)/(1-x^32)^2 +(x^32+x^96)/(1-x^64)^2 +(x^64+x^192)/(1-x^128)^2 +(x^128+x^256+x^384)/(1-x^256)^2. - Robert Israel, Aug 26 2016

a(n) = 2*a(n-256) - a(n-512). - Charles R Greathouse IV, Aug 26 2016

From Peter Bala, Feb 27 2019: (Start)

a(n) = numerator(n/(n + 256)).

O.g.f.: F(x) - Sum_{k = 1..8} F(x^(2^k)), where F(x) = x/(1 - x)^2. Cf. A106617. (End)

From Amiram Eldar, Nov 26 2022: (Start)

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) - 1/2^(6*s) - 1/2^(7*s) - 1/2^(8*s)).

Multiplicative with a(2^e) = 2^(e-min(e,8)), and a(p^e) = p^e for p > 2.

Sum_{k=1..n} a(k) ~ (43691/131072) * n^2. (End)

Extensions

Keyword:mult added and terms a(51) and beyond from Andrew Howroyd, Jul 23 2018