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%I #49 Apr 19 2023 19:11:14
%S 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,
%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 53,27,55,7,57,29,59,15,61,31,63,2,65,33,67,17,69,35
%N a(n) = n/gcd(n,2^5), n >= 0.
%C 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).
%C 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.
%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="/A227140/b227140.txt">Table of n, a(n) for n = 0..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_64">Index entries for linear recurrences with constant coefficients</a>, 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).
%F a(n) = n/gcd(n, 2^5).
%F a(n) = denominator(8*(n-4)/n), n >= 0 (with denominator(infinity) = 0).
%F From _Peter Bala_, Feb 27 2019: (Start)
%F a(n) = numerator(n/(n + 32)).
%F 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)
%F From _Bernard Schott_, Mar 02 2019: (Start)
%F a(n) = 1 iff n is 1, 2, 4, 8, 16, 32 and a(2^n) = 2^(n-5) for n >= 5.
%F a(n) = n iff n is odd (A005408). (End)
%F From _Amiram Eldar_, Nov 25 2022: (Start)
%F Multiplicative with a(2^e) = 2^(e-min(e,5)), and a(p^e) = p^e for p > 2.
%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)).
%F Sum_{k=1..n} a(k) ~ (683/2048) * n^2. (End)
%p seq(n/igcd(n,32),n=0..70); # _Muniru A Asiru_, Feb 28 2019
%t With[{c=2^5},Table[n/GCD[n,c],{n,0,70}]] (* _Harvey P. Dale_, Feb 16 2018 *)
%o (PARI) a(n)=n/gcd(n, 2^5); \\ _Andrew Howroyd_, Jul 23 2018
%o (Magma) [n/GCD(n, 2^5): n in [0..80]]; // _G. C. Greubel_, Feb 27 2019
%o (Sage) [n/gcd(n,2^5) for n in (0..80)] # _G. C. Greubel_, Feb 27 2019
%o (GAP) List([0..80], n-> n/Gcd(n, 2^5)); # _G. C. Greubel_, Feb 27 2019
%Y Cf. A000265, A227042, A227107, A106617, A276234.
%K nonn,frac,easy,mult
%O 0,4
%A _Wolfdieter Lang_, Jul 04 2013
%E Keyword:mult added by _Andrew Howroyd_, Jul 23 2018
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