

A227140


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


4



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
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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(n4, 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.


FORMULA

a(n) = n/gcd(n, 2^5).
a(n) = denominator(8*(n4)/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^(n5) 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^(emin(e,5)), and a(p^e) = p^e for p > 2.
Dirichlet g.f.: zeta(s1)*(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,changed


AUTHOR

Wolfdieter Lang, Jul 04 2013


EXTENSIONS

Keyword:mult added by Andrew Howroyd, Jul 23 2018


STATUS

approved



