OFFSET
1,4
COMMENTS
In other words: a(n) is the number of compositions of the n-th odd number into primes of form x^2 + y^2.
Note that a(4)..a(10) = [2, 3, 4, 6, 9, 15, 24] is also the number of laps related to the orbital resonances of the seven Earth-sized planets [h, g, f, e, d, c, b] in the planetary system of the TRAPPIST-1 star (see links). Note also that Lcm(2,3,4,6,9,15,24) = 2^3*3^2*5^1 = 8*9*5 = 360.
LINKS
Matthew S. Clement, Sean N. Raymond, Dimitri Veras, and David Kipping, Mathematical encoding within multi-resonant planetary systems as SETI beacons, arXiv:2204.14259 [astro-ph.EP], (2022), see page 6.
Michaël Gillon, Emmanuël Jehin, et al., Earth-sized planets transiting a nearby ultracool dwarf star, Nature (02 May 2016), doi: 10.1038/nature17448.
Michaël Gillon, A. Triaud, et al., Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1, arXiv:1703.01424 [astro-ph.EP], 2017; Nature 542, 456-460 (2017).
NASA, Jet Propulsion Laboratory, California Institute of Technology, NASA & TRAPPIST-1: A Treasure Trove of Planets Found, Youtube video (2017).
NASA, Jet Propulsion Laboratory, California Institute of Technology, SPITZER Space Telescope, TRAPPIST-1
V. Pletser and L. Basano, Exponential distance relation and near resonances in the Trappist-1 planetary system, arXiv:1703.04545 [astro-ph.IM] (2017).
Daniel Tamayo, Hanno Rein, Cristobal Petrovich, and Norman Murray, Convergent Migration Renders TRAPPIST-1 Long-lived, arXiv:1704.02957v2 [astro-ph.EP], The Astrophysical Journal Letters, Volume 840, Issue 2, article id. L19, 6 pp. (2017).
D. Tamayo, M. Russo, and A. Santaguida, The song of a solar system: TRAPPIST-1, Youtube (2017), see from minute 1:10.
Dan Tepfer, TRAPPIST-1 in musical notation
TRAPPIST-1, TRAPPIST-1
Wikipedia, TRAPPIST-1
EXAMPLE
For n = 8 we have that 2*8 - 1 = 15, and the elements of A002313 that are <= 15 are [2, 5, 13], and the compositions of 15 that contain only some of these three prime numbers are [13,2], [2,13], [5,5,5], [5,2,2,2,2,2], [2,5,2,2,2,2], [2,2,5,2,2,2], [2,2,2,5,2,2], [2,2,2,2,5,2], [2,2,2,2,2,5], there are 9 such compositions so a(8) = 9. - Omar E. Pol, May 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, May 20 2017
EXTENSIONS
a(31)-a(41) from Alois P. Heinz, May 19 2022
Partially edited by N. J. A. Sloane, Dec 04 2023
STATUS
approved