A111728 -id:A111728 - cp's OEIS frontend

A236258 Decimal expansion of 2 + 21/4*(4/11)^(4/3).

3, 3, 6, 2, 6, 4, 3, 9, 0, 5, 9, 6, 1, 4, 3, 3, 7, 8, 0, 3, 7, 3, 6, 2, 7, 2, 5, 7, 0, 0, 1, 4, 4, 4, 1, 9, 9, 9, 4, 6, 0, 6, 6, 1, 3, 6, 3, 0, 6, 3, 4, 5, 4, 0, 0, 4, 7, 5, 2, 8, 7, 4, 3, 5, 7, 9, 7, 8, 4, 0, 5, 5, 3, 4, 9, 2, 9, 1, 7, 6, 2, 5, 9, 7, 5, 2, 7, 7, 0, 1, 2, 5, 9, 7, 9, 6, 6, 5, 0, 9, 6, 5, 5, 7, 9

Offset: 1

Arkadiusz Wesolowski, Jan 21 2014

Evolution of the effective number of relativistic degrees of freedom contributing to energy density, g(*), can be seen on a graph as a function of temperature. At the energy scales below 0.1 MeV, g(*) is equal to this constant (in the Standard Model and in the minimal extension of Standard Model).

			3.362643905961433780373627257001444199946066136306345400475287435797840...

Benjamin Bederson, More Things in Heaven and Earth: A Celebration of Physics at the Millennium, Springer-Verlag, New York, 1999, p. 272.
J. C. Niemeyer and J. W. Truran, Type la Supernovae: Theory and Cosmology, Cambridge University Press, 2000, p. 107.

Hannu Kurki-Suonio, Cosmology I, Chapter 6: Thermal history of the Early Universe, p. 64.

Cf. A111728.

n:=2+21/4*(RealField(105)!4/11)^(4/3); Reverse(Intseq(Floor(10^104*n)));

Digits:=100: evalf(2+21/4*(4/11)^(4/3)); # Wesley Ivan Hurt, Oct 05 2014

RealDigits[N[2 + 21/4*(4/11)^(4/3), 105]][[1]]

default(realprecision, 105); x=2+21/4*(4/11)^(4/3); for(n=1, 105, d=floor(x); x=(x-d)*10; print1(d, ", "));

A225357 Decimal expansion of (4/7)*(43/57)^(4/3).

Original entry on oeis.org

3, 9, 2, 4, 2, 2, 0, 6, 4, 1, 6, 0, 8, 3, 2, 7, 3, 2, 3, 4, 0, 8, 7, 9, 4, 2, 0, 3, 6, 1, 2, 6, 4, 4, 4, 1, 3, 7, 8, 2, 4, 0, 8, 8, 5, 5, 1, 1, 1, 3, 3, 9, 2, 5, 6, 2, 8, 4, 8, 8, 6, 5, 4, 9, 9, 5, 4, 8, 1, 9, 9, 0, 2, 7, 3, 3, 1, 4, 8, 7, 2, 7, 0, 0, 1, 6, 0, 8, 5, 8, 7, 3, 5, 2, 9, 8, 5, 0, 1, 0, 6, 9, 3, 6, 2

Offset: 0

Omar E. Pol, May 28 2013

Steven Weinberg suggested that the contribution from Goldstone bosons to the effective number of neutrino species present in the era before recombination would be (4/7)*(43/57)^(4/3).

			0.3924220641608327323408794203612644413782408855111339...

Steven Weinberg, Goldstone bosons as fractional cosmic neutrinos, arXiv:1305.1971 [astro-ph.CO], 2013.

Cf. A111728, A225359.

RealDigits[(4/7)*(43/57)^(4/3), 10, 110][[1]] (* T. D. Noe, Jun 03 2013 *)

numerical_approx((4/7)*(43/57)^(4/3), digits=115) # G. C. Greubel, Mar 19 2022

Equals decimal expansion of 172*(43/57)^(1/3)/399.

A383267 Decimal expansion of (4/11)^(1/3).

Original entry on oeis.org

7, 1, 3, 7, 6, 5, 8, 5, 5, 5, 0, 3, 6, 0, 8, 1, 7, 0, 6, 7, 1, 8, 9, 9, 9, 9, 1, 7, 6, 2, 6, 6, 1, 2, 4, 7, 5, 9, 0, 7, 9, 6, 5, 4, 7, 5, 8, 9, 0, 3, 8, 0, 6, 6, 9, 1, 5, 6, 2, 6, 7, 5, 2, 0, 8, 4, 5, 8, 3, 1, 4, 7, 0, 6, 7, 7, 1, 8, 7, 5, 6, 4, 6, 3, 2, 4, 0, 3, 3, 9, 9, 3, 2, 2, 6, 8, 1, 7, 1, 7, 2, 4, 4, 6, 4

Offset: 0

Arkadiusz Wesolowski, Apr 21 2025

In the standard cosmology, the temperature of the free-streaming neutrinos which formed the cosmic neutrino background is (4/11)^(1/3) of the relic photon temperature after the electron-positron annihilation in the early universe (assuming that all electrons and positrons annihilated into photons).

			0.713765855503608170671899991762661247590796547589038066915626752084583...

E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley, Redwood City, CA, 1990, p. 503 Appendix A.
R. E. Lopez, S. Dodelson, A. Heckler and M. S. Turner, Precision detection of the cosmic neutrino background, Physical Review Letters 82 (1999) 3952-3955, p. 3952.
Steven Weinberg, Gravitation and Cosmology Principles and Applications of the General Theory of Relativity, John Wiley, New York, 1972, p. 537.

Wikipedia, Cosmic neutrino background.

Cf. A111728.

SetDefaultRealField(RealField(106)); n:=(4/11)^(1/3); Reverse(Intseq(Floor(10^105*n)));

RealDigits[(4/11)^(1/3),10,105][[1]] (* Stefano Spezia, Apr 25 2025 *)

PARI
```
(4/11)^(1/3)
```

Equals 1/A111728 = A005480/A010583.

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A236258 Decimal expansion of 2 + 21/4*(4/11)^(4/3).

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A225357 Decimal expansion of (4/7)*(43/57)^(4/3).

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A383267 Decimal expansion of (4/11)^(1/3).

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