A287148 Number of compositions (ordered partitions) of 2*n-1 into primes of form x^2 + y^2.

0, 0, 1, 2, 3, 4, 6, 9, 15, 24, 37, 58, 92, 149, 243, 393, 629, 1004, 1603, 2564, 4106, 6571, 10508, 16807, 26895, 43060, 68952, 110392, 176696, 282798, 452616, 724441, 1159537, 1855919, 2970476, 4754382, 7609712, 12180021, 19495286, 31203935, 49944397

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

Table of n, a(n) for n=1..41.

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

Bisection of A282971.

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

Sequence in context: A081237 A022159 A022161 * A127604 A268710 A223900

Adjacent sequences: A287145 A287146 A287147 * A287149 A287150 A287151

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