A276234 -id:A276234 - cp's OEIS frontend

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

Wolfdieter Lang, Jul 04 2013

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

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).

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

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

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

seq(n/igcd(n,32),n=0..70); # Muniru A Asiru, Feb 28 2019

With[{c=2^5},Table[n/GCD[n,c],{n,0,70}]] (* Harvey P. Dale, Feb 16 2018 *)

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

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

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)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A276233 a(n) = (n+256)/gcd(n,256).

Original entry on oeis.org

257, 129, 259, 65, 261, 131, 263, 33, 265, 133, 267, 67, 269, 135, 271, 17, 273, 137, 275, 69, 277, 139, 279, 35, 281, 141, 283, 71, 285, 143, 287, 9, 289, 145, 291, 73, 293, 147, 295, 37, 297, 149, 299, 75, 301, 151, 303, 19, 305, 153

Offset: 1

Artur Jasinski, Aug 24 2016

nonn

Ray Chandler, Table of n, a(n) for n = 1..1024
Index entries for linear recurrences with constant coefficients, order 512.

Cf. A276234 (denominators).

seq((n+256)/igcd(n,256),n=1..300); # Robert Israel, Aug 25 2016

Numerator[Table[Limit[EllipticTheta[3, 0, b]^8 + EllipticTheta[2, 0,Sqrt[b]]^8/(n b),b -> 0], {n, 1, 50}]]
Table[(n + 256)/GCD[n, 256], {n, 60}] (* Ray Chandler, Aug 03 2023 *)

a(n) = numerator of 1+256/n, which is the limit of the function EllipticTheta(3, 0, q)^8 + EllipticTheta(2, 0, sqrt(q))^8/(n q) when q -> 0.
a(2k-1) = n + 256 = 2k-1 + 256 = 2k + 255
a(4k-2) = n/2 + 128 = 2k-1 + 128 = 2k + 127
a(8k-4) = n/4 + 64 = 2k-1 + 64 = 2k + 63
a(16k-8) = n/8 + 32 = 2k-1 + 32 = 2k + 31
a(32k-16) = n/16 + 16 = 2k-1 + 16 = 2k + 15
a(64k-32) = n/32 + 8 = 2k-1 + 8 = 2k + 7
a(128k-64) = n/64 + 4 = 2k-1 + 4 = 2k + 3
a(256k-128) = n/128 + 2 = 2k-1 + 2 = 2k + 1.
a(n) = 2*a(n-256) - a(n-512) for n > 512. - Ray Chandler, Aug 03 2023

cp's OEIS Frontend

A227140 a(n) = n/gcd(n,2^5), n >= 0.

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