Lucky odd numbers
The widely held belief that odd numbers were luckier than even ones. Roman writers explained the 355 days of Numa's year by it.
How the Romans fixed the year
Censorinus, On the Birthday 20–22 (238 CE), in the edition of Friedrich Hultsch (Leipzig, 1867) · 238 CE
postea sive a Numa, ut ait Fulvius, sive, ut Iunius, a Tarquinio XII facti sunt menses et dies CCCLV, quamvis luna XII suis mensibus CCCLIIII dies videbatur explere. sed ut dies unus abundaret, aut per inprudentiam accidit, aut, quod magis credo, ea superstitione qua inpar numerus plenus et magis faustus habebatur.
Later there came to be twelve months and 355 days. Fulvius says that Numa did this, and Junius that Tarquin did. Yet the moon seemed to complete its twelve months in 354 days. The one extra day was either a mistake or, as I rather believe, came from the superstition that held an odd number to be full and luckier.
certe ad annum priorem unus et quinquaginta dies accesserunt: qui quia menses duo non explerent, sex illis cavis mensibus dies sunt singuli detracti et ad eos additi, factique dies LVII, et ex his duo menses, Ianuarius undetriginta dierum, Februarius duodetriginta. adque ita omnes menses pleni et inpari dierum numero esse coeperunt, excepto Februario, qui solus cavus et ob hoc ceteris infaustior est habitus.
In any case, 51 days were added to the old year. These were not enough to fill two months, so one day was taken from each of the six hollow months and added to them. That made 57 days, and from these came two months, January with 29 days and February with 28. So all the months came to be full, with an odd number of days, except February. It alone was hollow, and so it was thought less lucky than the rest.
Where the months got their names
Macrobius, Saturnalia 1.12–14 (early fifth century CE), in the edition of Franz Eyssenhardt (Leipzig, 1893) · early fifth century CE; the dialogue is set in the 380s
paulo post Numa in honorem imparis numeri, secretum hoc et ante Pythagoram parturiente natura, unum adiecit diem, quem Ianuario dedit, ut tam in anno quam in mensibus singulis praeter unum Februarium impar numerus seruaretur. nam quia duodecim menses, si singuli aut pari aut impari numero putarentur, consummationem parem facerent, unus pari numero institutus uniuersam putationem imparem fecit.
A little later Numa added one more day in honor of the odd number, a secret that nature revealed even before Pythagoras. He gave the day to January, so that the year and every month except February would have an odd number of days. If all twelve months had an even number of days, or all had an odd number, the total would be even. So one month with an even number made the whole count odd.