A227140 -id:A227140 - cp's OEIS frontend

A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

1, 3, 1, 2, 5, 1, 5, 3, 7, 1, 3, 7, 4, 9, 1, 7, 1, 1, 5, 11, 1, 4, 9, 5, 11, 6, 13, 1, 9, 5, 11, 3, 13, 7, 15, 1, 5, 11, 2, 13, 7, 5, 8, 17, 1, 11, 3, 13, 7, 3, 2, 17, 9, 19, 1, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 1

Offset: 1

Wolfdieter Lang, Jul 01 2013

See the comments under A227041. a(n,m) gives the denominator of H(n,m) = 2*n*m/(n+m) in lowest terms.

			The triangle of denominators of H(n,m), called a(n,m) begins:
n\m  1   2   3   4   5    6    7    8    9   10  11 ...
1:   1
2:   3   1
3:   2   5   1
4:   5   3   7   1
5:   3   7   4   9   1
6:   7   1   1   5  11    1
7:   4   9   5  11   6   13    1
8;   9   5  11   3  13    7   15    1
9:   5  11   2  13   7    5    8   17    1
10: 11   3  13   7   3    2   17    9   19    1
11:  6  13   7  15   8   17    9   19   10   21   1
...
For the triangle of the rationals H(n,m) see the example section of A227041.
H(4,2) = denominator(16/6) = denominator(8/3) = 3 = 6/gcd(6,8) = 6/2.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Cf. A227041, A026741 (column m=1), A000265 (m=2), A106619 (m=3), A227140(n+8) (m=4), A227108 (m=5), A221918/A221919.

a(n,m) = denominator(2*n*m/(n+m)), 1 <= m <= n.

a(n,m) = (n+m)/gcd(2*n*m, n+m) = (n+m)/gcd(n+m, 2*m^2), 1 <= m <= n.

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

Original entry on oeis.org

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

Artur Jasinski, Aug 24 2016

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

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.

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

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

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

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

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

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

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

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)

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

A227107 Numerators of harmonic mean H(n,4), n >= 0.

Original entry on oeis.org

0, 8, 8, 24, 4, 40, 24, 56, 16, 72, 40, 88, 6, 104, 56, 120, 32, 136, 72, 152, 20, 168, 88, 184, 48, 200, 104, 216, 7, 232, 120, 248, 64, 264, 136, 280, 36, 296, 152, 312, 80, 328, 168, 344, 22, 360, 184, 376, 96, 392, 200, 408, 52, 424, 216

Offset: 0

Wolfdieter Lang, Jul 01 2013

a(n) = numerator(H(n,4)) = numerator(8*n/(n+4)), n>=0, with H(n,4) the harmonic mean of n and 4.
The corresponding denominators are given in A000265(n+4), n >= 0.
a(n+4), n>=0, is the fourth column (m=4) of the triangle A227041.

			The rationals H(n,4) begin: 0, 8/5, 8/3, 24/7, 4, 40/9, 24/5, 56/11, 16/3, 72/13, 40/7, 88/15, 6, 104/17, 56/9, 120/19, ...

Cf. A227041(n+4,4), A227140(n+8) (denominators), n >= 0.

a(n) = numerator(8*n/(n+4)), n >= 0.

a(n) = 8*n/gcd(n+4,8*n) = 8*n/gcd(n+4,32), n >= 0.

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A227042 Triangle of denominators of harmonic mean of n and m, 1 <= m <= n.

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

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A227107 Numerators of harmonic mean H(n,4), n >= 0.

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