|
%I #46 Nov 26 2022 04:03:11
%S 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,
%T 29,15,31,1,33,17,35,9,37,19,39,5,41,21,43,11,45,23,47,3,49,25,51,13,
%U 53,27,55,7,57,29,59,15,61,31,63,1,65,33,67,17,69,35,71
%N a(n) = n/gcd(n, 256).
%C a(n) first differs from A000265(n) at n = 512. - _Andrew Howroyd_, Jul 23 2018
%C 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
%H Andrew Howroyd, <a href="/A276234/b276234.txt">Table of n, a(n) for n = 1..1000</a>
%H Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function</a>, 2019.
%H <a href="/index/Rec#order_512">Index entries for linear recurrences with constant coefficients</a>, order 512.
%F a(2k-1) = 2k-1.
%F 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
%F a(n) = 2*a(n-256) - a(n-512). - _Charles R Greathouse IV_, Aug 26 2016
%F From _Peter Bala_, Feb 27 2019: (Start)
%F a(n) = numerator(n/(n + 256)).
%F O.g.f.: F(x) - Sum_{k = 1..8} F(x^(2^k)), where F(x) = x/(1 - x)^2. Cf. A106617. (End)
%F From _Amiram Eldar_, Nov 26 2022: (Start)
%F 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)).
%F Multiplicative with a(2^e) = 2^(e-min(e,8)), and a(p^e) = p^e for p > 2.
%F Sum_{k=1..n} a(k) ~ (43691/131072) * n^2. (End)
%p seq(n/igcd(n,256), n=1..100); # _Robert Israel_, Aug 26 2016
%t Table[n/GCD[n, 2^8], {n,1,80}] (* _G. C. Greubel_, Feb 27 2019 *)
%o (PARI) a(n)=n/gcd(n,256) \\ _Charles R Greathouse IV_, Aug 26 2016
%o (Magma) [n/GCD(n, 2^8): n in [1..80]]; // _G. C. Greubel_, Feb 27 2019
%o (Sage) [n/gcd(n, 2^8) for n in (1..80)] # _G. C. Greubel_, Feb 27 2019
%o (GAP) List([1..80], n-> n/Gcd(n, 2^8)); # _G. C. Greubel_, Feb 27 2019
%Y Cf. A276233 (numerators), A227140, A000265, A106617.
%K nonn,easy,mult
%O 1,3
%A _Artur Jasinski_, Aug 24 2016
%E Keyword:mult added and terms a(51) and beyond from _Andrew Howroyd_, Jul 23 2018
|
