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Given the existence of large star catalogues listing millions of astronomical objects with numerical designations (such as Tabby's Star, KIC 8462852) it occurred to me that one of them might contain the Sun. However, I had no luck.

I searched NASA's HEASARC which combines GRB, SIMBAD, VizieR, NED, and BSC.

HEASARC found Sirius in SIMBAD, NED, and BSC (HR 2491), but there were no results for "Sun" or "Sol".

So, is there any catalogue (maybe a stellar evolution catalogue) that puts the Sun on equal footing with other stars, giving it a canonical number designation?

Bonus high fives if that number is 0 or 1. (Sorry, HR1 is a 6.71 magnitude star in Andromeda, also known as HD 3).

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    $\begingroup$ JPL HORIZONS has the Target Body=10 as the Sun [Sol] [10] but since it doesn't have any other stars in there so I don't think it meets the criteria of your question. The CALSPEC database has the Sun along with many other spectrophotometric standards but none of the objects in that database are numbered (other than their original catalog designation such as HD xyz, BD xyz etc) so don't think this meets the question either $\endgroup$ – astrosnapper Sep 14 '19 at 18:31
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It's somewhat difficult to prove it either way given the large number of catalogues out there. The majority (at least of the ones on VizieR) do not include the Sun.

The only case I'm aware of that almost meets your requirements is in the Preliminary Version of the Third Catalogue of Nearby Stars by Gliese and Jahreiß (1991). This catalogue is well-known as the source of stellar designations beginning "Gliese" and "GJ", and is queryable via VizieR.

The first record in the catalogue is the Sun, which includes its spectral type, apparent magnitude, colour indices and space velocity as for other stars. Right ascension, declination and proper motion aren't included for obvious reasons. It doesn't match your requirement for having a numerical designation because the name field is just "Sun", while the other stars in the catalogue have numbers. The "NN" stars originally didn't have numbers either but these were added in 1997: in the literature these objects are still typically referred to as "GJ ####" rather than "NN ####".

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