A348131 -id:A348131 - cp's OEIS frontend

A348132 a(n) is the denominator of the relativistic sum of n velocities of 1/n, in units where the speed of light is 1.

1, 5, 9, 353, 275, 66637, 18571, 24405761, 2215269, 14712104501, 411625181, 13218256749601, 109949704423, 16565151205544957, 39931933598775, 27614800115689879553, 18928981513351817, 59095217374989483261925, 11350851717672992089, 157904201452248753415276001

Offset: 1

Amiram Eldar, Oct 01 2021

			For n = 2, the sum of two velocities of 1/2 is (1/2 + 1/2)/(1 + (1/2)*(1/2)) = 4/5, thus a(2) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..387
Wikipedia, Velocity-addition formula.

Cf. A348051, A348052, A348131 (numerators).

f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 + s/n), {k, 1, n - 1}]; s]; Denominator @ Array[f, 20]

a(2n-1) = n^(2n-1) + (n-1)^(2n-1) and a(2n) = ((2n+1)^(2n) + (2n-1)^(2n)) / 2. - Thomas Ordowski, Feb 12 2022

A348051 Triangle T(j,k) of numerators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

Original entry on oeis.org

4, 5, 3, 7, 9, 12, 2, 7, 11, 8, 3, 11, 16, 13, 20, 11, 7, 19, 17, 25, 15, 10, 13, 17, 16, 23, 27, 24, 7, 1, 13, 3, 5, 5, 19, 5, 11, 13, 4, 1, 8, 31, 29, 17, 28, 14, 17, 5, 4, 31, 39, 36, 23, 37, 48, 13, 2, 23, 19, 29, 9, 33, 11, 7, 9, 21, 5, 19, 26, 23, 34, 41, 37, 9, 14, 53, 49, 56

Offset: 2

Hugo Pfoertner, Sep 25 2021

The velocities are assumed to be given in units of c, and thus c = 1.

			The triangle of added fractions begins:
     u   1/2   1/3   2/3   1/4   2/5   3/5   3/4   1/5   2/7   3/8   3/7
   v \    .     .     .     .     .     .     .     .     .     .     .
  1/2 |  4/5    .     .     .     .     .     .     .     .     .     .
  1/3 |  5/7   3/5    .     .     .     .     .     .     .     .     .
  2/3 |  7/8   9/11 12/13   .     .     .     .     .     .     .     .
  1/4 |  2/3   7/13 11/14  8/17   .     .     .     .     .     .     .
  2/5 |  3/4  11/17 16/19 13/22 20/29   .     .     .     .     .     .
  3/5 | 11/13  7/9  19/21 17/23 25/31 15/17   .     .     .     .     .
  3/4 | 10/11 13/15 17/18 16/19 23/26 27/29 24/25   .     .     .     .
  1/5 |  7/11  1/2  13/17  3/7   5/9   5/7  19/23  5/13   .     .     .
  2/7 | 11/16 13/23  4/5   1/2   8/13 31/41 29/34 17/37 28/53   .     .
  3/8 | 14/19 17/27  5/6   4/7  31/46 39/49 36/41 23/43 37/62 48/73   .
  3/7 | 13/17  2/3  23/27 19/31 29/41  9/11 33/37 11/19  7/11  9/13 21/29

Wikipedia, Velocity-addition formula.

A348052 are the corresponding denominators.

Cf. A007305, A007306, A348131, A348132.

A348052 Triangle T(j,k) of denominators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

Original entry on oeis.org

5, 7, 5, 8, 11, 13, 3, 13, 14, 17, 4, 17, 19, 22, 29, 13, 9, 21, 23, 31, 17, 11, 15, 18, 19, 26, 29, 25, 11, 2, 17, 7, 9, 7, 23, 13, 16, 23, 5, 2, 13, 41, 34, 37, 53, 19, 27, 6, 7, 46, 49, 41, 43, 62, 73, 17, 3, 27, 31, 41, 11, 37, 19, 11, 13, 29, 6, 25, 29, 32, 43, 47, 40, 13, 19, 68, 61, 65

Offset: 2

Hugo Pfoertner, Sep 25 2021

			See A348051.
The triangle starts:
  4/5,
  5/7,  3/5,
  7/8,  9/11, 12/13,
  2/3,  7/13, 11/14,  8/17,
  3/4, 11/17, 16/19, 13/22, 20/29
....

A348051 are the corresponding numerators.

Cf. A007305, A007306, A348131, A348132.

A348829 Numerator of relativistic sum w(2n) of the velocities v = 1/p^(2n) over all primes p, in units where the speed of light c = 1.

Original entry on oeis.org

3, 1, 12, 59, 521, 872492, 415603, 471263387, 100453109125251, 249063001217323, 1206701295264057, 2340564635396243082668, 1836709980831869650909, 7917057291763619291770993, 6790679763108188972468718224386027, 497252110757159525928442098399943

Offset: 1

Thomas Ordowski, Nov 01 2021

Generally, for a complex number s, w(s) = tanh(Sum_{p prime} arctanh(1/p^s)), assuming that Re(s) > 1.
Theorem. If Re(s) > 1, then w(s) = (1 - t(s))/(1 + t(s)) with t(s) = zeta(2s)/zeta(s)^2, where zeta(z) is the Riemann zeta function of z.
Proof. Einstein's formula w = (u + v)/(1 + uv) can be expanded as (1-w)/(1+w) = ((1-u)/(1+u))((1-v)/(1+v))... for any number of velocities u, v, ... Hence, by the Euler product, Product_{p prime} (1-1/p^s)/(1+1/p^s) = zeta(2s)/zeta(s)^2, qed. Note that the function f(x) = (1-x)/(1+x) is an involution.
If an integer s > 0 is even, then w(s) is rational (related to the Bernoulli numbers B_{s} and B_{2s}).
Conjecture: if an odd integer s > 1, then w(s) is irrational. Cf. W. Kohnen (link).
Note: Apery's constant zeta(3) = 1.202... is irrational.

			w(2) = 3/7, w(4) = 1/13, w(6) = 12/703, ...

Winfried Kohnen, Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 99, No. 3 (1989), pp. 231-233.
Wikipedia, Euler product.
Wikipedia, Riemann zeta function.
Wikipedia, Velocity-addition formula.

The denominators are A348830.
Cf. A114362, A114363.
See also A348131, A348132.

r[s_] := Zeta[2*s]/Zeta[s]^2; w[s_] := (1 - r[s])/(1 + r[s]); Table[Numerator[w[2*n]], {n, 1, 15}] (* Amiram Eldar, Nov 01 2021 *)

a(n) = Numerator(tanh(Sum_{p prime} arctanh(1/p^(2n)))).
a(n) = Numerator((zeta(2n)^2-zeta(4n))/(zeta(2n)^2+zeta(4n))).
a(n) = Numerator((1-t(2n))/(1+t(2n))), where t(2n) = A114362(n)/A114363(n).
If Re(s) > 1, then w(s) = f(f(w(s))) = (1-t(s))/(1+t(s)) and t(s) = f(f(t(s))) = (1-w(s))/(1+w(s)) = zeta(2s)/zeta(s)^2, where f(x) = (1-x)/(1+x). See my theorem and the note under my proof of this theorem. - Thomas Ordowski, Jan 03 2022
Conjecture: 0 < w(2n) - (1/2^(2n) + 1/3^(2n) + 1/5^(2n) + 1/7^(2n)) < 1/11^(2n) for every n > 0. Amiram Eldar confirmed my conjecture numerically up to n = 10^4. - Thomas Ordowski, Nov 13 2022
It can be proven that P(2n) - w(2n) ~ 1/12^(2n), where P(x) = Sum_{prime p} 1/p^x = 1/2^x + 1/3^x + 1/5^x + ... is the prime zeta function of real x > 1. - Thomas Ordowski, Nov 06 2024

More terms from Amiram Eldar, Nov 01 2021

A348140 a(n) is the numerator of tan(n * arctan(1/n)).

Original entry on oeis.org

1, 4, 13, 240, 719, 42372, 92567, 14970816, 21475201, 8825080100, 7836127861, 7809130867824, 4132643140079, 9678967816041188, 2973238691433583, 16000787866533953280, 2798084251807349761, 34017524842099233036996, 3336132453587291393821, 90417110945911655996319600

Offset: 1

Amiram Eldar, Oct 02 2021

			The fractions begin with 1, 4/3, 13/9, 240/161, 719/475, 42372/27755, 92567/60319, 14970816/9722113, 21475201/13913289, 8825080100/5707904499, ...

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A049471, A348131, A348132, A348141 (denominators).

f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 - s/n), {k, 1, n - 1}]; s]; Numerator @ Array[f, 20]

Lim_{n->oo} a(n)/A348141(n) = tan(1) (A049471).

A348141 a(n) is the denominator of tan(n * arctan(1/n)).

Original entry on oeis.org

1, 3, 9, 161, 475, 27755, 60319, 9722113, 13913289, 5707904499, 5061910249, 5039646554593, 2665025213747, 6237995487261915, 1915304790146175, 10303367499652761601, 1801181048868783377, 21891769059478538933603, 2146451844926024801801, 58162468900440912135124001

Offset: 1

Amiram Eldar, Oct 02 2021

Amiram Eldar, Table of n, a(n) for n = 1..387

Cf. A348131, A348132, A348140 (numerators).

f[n_] := Module[{s = 1/n}, Do[s = (s + 1/n)/(1 - s/n), {k, 1, n - 1}]; s]; Denominator @ Array[f, 20]

cp's OEIS Frontend

A348132 a(n) is the denominator of the relativistic sum of n velocities of 1/n, in units where the speed of light is 1.

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A348051 Triangle T(j,k) of numerators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

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A348052 Triangle T(j,k) of denominators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

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A348829 Numerator of relativistic sum w(2n) of the velocities v = 1/p^(2n) over all primes p, in units where the speed of light c = 1.

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A348140 a(n) is the numerator of tan(n * arctan(1/n)).

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A348141 a(n) is the denominator of tan(n * arctan(1/n)).

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Author

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Mathematica