A287148 - OEIS

%I #132 Dec 04 2023 11:15:38

%S 0,0,1,2,3,4,6,9,15,24,37,58,92,149,243,393,629,1004,1603,2564,4106,

%T 6571,10508,16807,26895,43060,68952,110392,176696,282798,452616,

%U 724441,1159537,1855919,2970476,4754382,7609712,12180021,19495286,31203935,49944397

%N Number of compositions (ordered partitions) of 2*n-1 into primes of form x^2 + y^2.

%C In other words: a(n) is the number of compositions of the n-th odd number into primes of form x^2 + y^2.

%C 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.

%H Matthew S. Clement, Sean N. Raymond, Dimitri Veras, and David Kipping, <a href="https://arxiv.org/abs/2204.14259">Mathematical encoding within multi-resonant planetary systems as SETI beacons</a>, arXiv:2204.14259 [astro-ph.EP], (2022), see page 6.

%H Michaël Gillon, Emmanuël Jehin, et al., <a href="http://www.trappist.one/files/Gillon_2016a.pdf">Earth-sized planets transiting a nearby ultracool dwarf star</a>, Nature (02 May 2016), doi: 10.1038/nature17448.

%H Michaël Gillon, A. Triaud, et al., <a href="https://arxiv.org/abs/1703.01424">Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1</a>, arXiv:1703.01424 [astro-ph.EP], 2017; Nature 542, 456-460 (2017).

%H NASA, Jet Propulsion Laboratory, California Institute of Technology, <a href="https://www.youtube.com/watch?v=bnKFaAS30X8">NASA & TRAPPIST-1: A Treasure Trove of Planets Found</a>, Youtube video (2017).

%H NASA, Jet Propulsion Laboratory, California Institute of Technology, SPITZER Space Telescope, <a href="https://www.spitzer.caltech.edu/explore/trappist-1">TRAPPIST-1</a>

%H V. Pletser and L. Basano, <a href="http://arxiv.org/abs/1703.04545">Exponential distance relation and near resonances in the Trappist-1 planetary system</a>, arXiv:1703.04545 [astro-ph.IM] (2017).

%H Daniel Tamayo, Hanno Rein, Cristobal Petrovich, and Norman Murray, <a href="https://arxiv.org/abs/1704.02957">Convergent Migration Renders TRAPPIST-1 Long-lived</a>, arXiv:1704.02957v2 [astro-ph.EP], The Astrophysical Journal Letters, Volume 840, Issue 2, article id. L19, 6 pp. (2017).

%H D. Tamayo, M. Russo, and A. Santaguida, <a href="https://www.youtube.com/watch?v=7i8Urhbd6eI">The song of a solar system: TRAPPIST-1</a>, Youtube (2017), see from minute 1:10.

%H Dan Tepfer, <a href="http://dantepfer.com/blog/?attachment_id=668">TRAPPIST-1 in musical notation</a>

%H TRAPPIST-1, <a href="http://trappist.one">TRAPPIST-1</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/TRAPPIST-1">TRAPPIST-1</a>

%e 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

%Y Bisection of A282971.

%Y Cf. A000040, A002313, A011782 (number of compositions of n).

%K nonn

%O 1,4

%A _Omar E. Pol_, May 20 2017

%E a(31)-a(41) from _Alois P. Heinz_, May 19 2022

%E Partially edited by _N. J. A. Sloane_, Dec 04 2023