Notable Properties of Specific Numbers

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20 is the 4^th tetrahedral number, the number of identical round objects (such as fruit in a produce stand) that it takes to build a tetrahedral stack with 4 levels and 4 along each edge of a triangular base. Here are the 4 layers, viewed from above; each O is an orange and the small dots show where the next layer is added:

o . o o o . o . . o o . o o o o . . o o . . . o o o (top) o o o o (bottom)

If an ordinary 3×3×3× Rubik's cube is in any legally accessible position (i.e. no impossible parity violations or moved stickers), then it can be transformed into any other valid position in no more than 20 moves, where 180-degree turns count as a single move. The problem of determining this number was posed by Morwen B. Thistlethwaite in 1981, and fully worked out in 2010 (see God's algorithm).

20 is the base of the ancient Mayan number system, one of the three known civilisations that independently developed a pure place-value system with a symbol for zero¹⁸. The Mayan names for the powers of 20 were:

kal = 20
bak = 20² = 400
pic = 20³ = 8000
calab = 20⁴ = 160000
kinchil = 20⁵ = 3200000
alau = 20⁶ = 64000000

See also 260 and 1872000.

20 as a "base" is also present some number names in many languages, for example: French (quatre-vingts, "four twenty" = 80), Danish (tre-sinds-tyve, "three times twenty" = 60), English (four score = 80), Basque (hirurogeita hamabost, "three-score-and ten-five" = 75), Ainu (tu hotnep, "two twenties" = 40), etc.

21 is the lowest base with 'easy' divisibility tests for 8 different numbers, assuming that the casting out 11's method is not considered 'easy'. In base 21 you can test for divisibility by 2, 3, 4, 5, 7, 10, 20 and 21.

Here is a list of record-setters for this property (bases with high numbers of testable divisors). The divisors in bold are tested just by looking at the last digit; those in the plain font are tested by the digit-addition technique (casting out 9's):

base	N	divisors
2	1	2
3	2	2, 3
4	3	2, 3, 4
6	4	2, 3, 5, 6
9	5	2, 3, 4, 8, 9
12	6	2, 3, 4, 6, 11, 12
16	7	2, 3, 4, 5, 8, 15, 16
21	8	2, 3, 4, 5, 7, 10, 20, 21
25	9	2, 3, 4, 5, 6, 8, 12, 24, 25
36	11	2, 3, 4, 5, 6, 7, 9, 12, 18, 35, 36
60	12	2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 59, 60
81	13	2, 3, 4, 5, 8, 9, 10, 16, 20, 27, 40, 80, 81
85	14	2, 3, 4, 5, 6, 7, 12, 14, 17, 21, 28, 42, 84, 85
120	18	2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 17, 20, 24, 30, 40, 60, 119, 120
225	19	2, 3, 4, 5, 7, 8, 9, 14, 15, 16, 25, 28, 32, 45, 56, 75, 112, 224, 225
240	20	2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 239, 240

Notice many lines that are all bold except for one or two terms: These are the highly composite numbers.

If you want the record-setters for prime divisors only, check the entry for 66. If you want to include casting out 11's as a divisibility test, see the entry for 29.

21 is sometimes whimsically called "eleventeen", but the word is more commonly used to refer to eleven (in reference to a child's age) or to an uncertain or indeterminate quantity, or to non-quantities such as a music album.

There are 22 ways to express 8 as a sum of positive integers: 8, 7+1, 6+2, 6+1+1, 5+3, 5+2+1, 5+1+1+1, 4+4, 4+3+1, etc. Such sums are called partitions because they represent ways to partition (divide or set apart) n objects from one another. The ways of partitioning 8 objects are illustrated graphically by the following diagrams (called Ferrers diagrams):

o o o o o o o o 8 , or 1+1+1+1+1+1+1+1 o o o o 4+2+1+1 o o o o o o o o o 7+1 , or 2+1+1+1+1+1+1 o o o o o o o o o 6+2 , or 2+2+1+1+1+1 o o o 3+3+2 o o o o o o o o o o o o o 6+1+1 , or 3+1+1+1+1+1 o o o o o 4+4 , or 2+2+2+2 o o o o o o o o o o 5+3 , or 2+2+2+1+1 o o o o 4+3+1 , or 3+2+2+1 o o o o o o o o o o o o 5+2+1 , or 3+2+1+1+1 o o o o o o 4+2+2 , or 3+3+1+1 o o o o o o o o o o 5+1+1+1 , or 4+1+1+1+1 o o o

The sequence (OEIS sequence A0041) begins: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, ...

This is one of the more important sequences in number theory; it has been studied deeply and has several mysteries. Ramanujan proved⁸³ the rather interesting fact that every 5^th term (starting with the 5) is divisible by 5 (for example, 5, 30 and 135), every 7^th term starting with the 7 is divisible by 7, and every 11^th term starting with the 11 is divisible by 11. There is a pattern involving divisibility by 13, but it is much more obscure: the 237^th term is 83561103925871, and every 17303^rd term thereafter is divisible by 13.

Smith numbers

22 is a Smith Number for base 10. A number is a Smith number if the sum of its digits equals the sum of the digits of its prime factors: 22 = 2 × 11 and 2 + 2 = 2 + 1 + 1. See also 4937775.

There are 22 letters of the Hebrew alphabet, in its simplest form. This is also the number of letters in the Aramaic and Phoenician alphabets. Hebrew has 27 letters if you include the five final forms, and 32 if you also count the five dual-phoneme letters twice (bet, vet, kaph, khaph, pe, phe, sin, shin, and tav/sav).

There are 22 trump cards in a standard Tarot deck. These are normally used as a trump suit in various card games, but are also important in the use of Tarot cards for divination, where they are called the Major Arcana. Older versions of Tarot differed, and the trumps disappear entirely when you trace it back far enough. (The Minchiate, my favourite version, has 40 trumps including 18 of the modern set, the 4 classical elements, 12 zodiac signs, and several more).

22.459157...

This is π^e. It is a little smaller than, and not quite as special as, e^π. See also e^π 3581.875516... and 11058015.34616.

22.92067661926415...

The normal harmonic series, the sum of all unit fractions 1/1 + 1/2 + 1/3 + 1/4 + ..., has an infinite sum. However, if you eliminate all fractions with a '9' digit in the denominator, you have the Kempner series, and the sum becomes finite. Aubrey J. Kempner proved (fairly easily) that the sum was less than 80, and Robert Baillie worked out an efficient way to compute the sum accurately.

The first prime that is a sum of three consecutive primes (5+7+11) (thanks to Philip Hassey for this tip)

The smallest number that requires more than two terms to express in the form X = 2^a3^b + 2^c3^d + 2^e3^f + ... See also 431

A cult number, noted particularly for associations with the Illuminati and conspiracy theories.

23.132741228718 = 8π-2

This is the simplest way to express "Gelfond's constant" e^π as a linear combination of integer multiples of 1 and π. If we subtract π to get 7π-2, then substutute 22/7 for π, the result is 20, showing that the original e^π is near 20 + π.

This coincidence has been explained in a more complicated way: by showing that (8π-2)/e^(πk²) is close to 1 when k=1 (because 8π-2 is 8πk²-2 when k=1), and then relating this to a series sum related to Jacobi Theta functions:

SUM_{(k=1..∞)}[(8πk²-2)e^(-πk²)] = 1

This works because the terms for k≥2 are all very small. However, this line of reasoning doesn't appear to predict any other near-integers, as is seen for example with many listed below that are of the form e^π√n.

23.140692632779269005...

(Gelfond's constant)

This is e^π, also called Gelfond's constant. It is notable because if you raise it to the power of a square root of a small positive integer, you often get an answer which is very near another (much larger) positive integer, or some other closed-form expression with square roots of integers. See 262537412640768743.999999... for more on this.

e^π also happens to be kind of close to 20 + π. The actual difference is 19.999099979189475767... Those two sets of 9's are the reason why there are two sets of digits that coincide:

23.140692632779269005... = e^π
3.141592653589793238... = π

As a result of this coincidence, there are approximations for e^π that are similar to approximations for π, such as the two mentioned in the entry for 1/7.

Values of (e^π)^√N = e^{π √N} fall into several categories, including near-integers, values near an integer times a power of phi, and various combinations involving square roots. Here is a fairly complete listing:

e^{π √5} = 2⁶(Φ)⁶ - 24   +   0.24... (W2)
e^{π √6} = 2⁶(1+√2)⁴ + 24   +   0.12... (W2)
e^{π √6} = (4√3)⁴ - 106   +   0.0091... (R2)
e^{π √10} = 2⁶(Φ)¹² + 24   +   0.013... (W2)
e^{π √10} = 12⁴ - 104   +   0.21... (R2)
e^{π √11} = 32³ + 738   +   0.14... (H1)
e^{π √13} = 2⁶((3+√13)/2)⁶ - 24   +   0.0033... (W2)
e^{π √13} = (12√2)⁴ + 104   +   0.052... (R2)
e^{π √14} = 4⁴(11+8√2)² - 104   +   0.034... (R4)
e^{π √15} = 3³(Φ)²(5+4√5)³ + 745   -   0.022... (H2)
e^{π √16} = 66³ - 744   -   0.68... (H1)
e^{π √19} = 96³ + 744   -   0.22... (H1)
e^{π √20} = 2³(25+13√5)³ - 744   +   0.15... (H2)
e^{π √22} = 2⁶(1+√12)¹² + 24   -   0.00011... (W2)
e^{π √22} = (12√11)⁴ - 104   +   0.0017... (R2)
e^{π √24} = 12³(1+√2)²(5+2√2)³ - 744   +   0.040... (H2)
e^{π √28} = 255³ - 744   -   0.011... (H1)
e^{π √30} = (4√3)⁴(5+4√2)⁴ - 104   +   0.00014... (R4)
e^{π √34} = 12⁴(4+√17)⁴ - 104   +   0.000048... (R4)
e^{π √35} = 16³(15+7√5)³ + 744   +   0.0016... (H2)
e^{π √37} = 2⁶(6+√37)⁶ - 24   +   0.0000013... (W2)
e^{π √37} = (84√2)⁴ + 104   +   0.000021... (R2)
e^{π √40} = 6³(65+27√5)³ - 744   +   0.00046... (H2)
e^{π √42} = 4⁴(21+8√6)⁴ - 104   -   0.0000062... (R4)
e^{π √43} = 960³ + 744   -   0.00022... (H1)
e^{π √46} = 12⁴(147+104√2)² - 104   +   0.0000024... (R4)
e^{π √51} = 48³(4+√17)²(5+√17)³ + 744   -   0.000035... (H2)
e^{π √52} = 30³(31+9√13)³ - 744   +   0.000028... (H2)
e^{π √58} = 2⁶((5+√29)/2)¹² + 24   -   0.000000011... (W2)
e^{π √58} = 396⁴ - 104   +   0.00000017... (R2)
e^{π √67} = 5280³ + 744   -   0.0000014... (H1)
e^{π √70} = (12√7)⁴(5√5+8√2)⁴ - 104   +   0.000000016... (R4)
e^{π √78} = (4√3)⁴(75+52√2)⁴ - 104   +   0.0000000038... (R4)
e^{π √82} = 12⁴(51+8√41)⁴ - 104   +   0.0000000019... (R4)
e^{π √88} = 60³(155+108√2)³ - 744   +   0.000000031... (H2)
e^{π √91} = 48³(227+63√13)³ + 744   +   0.000000019... (H2)
e^{π √102} = (4√3)⁴(200+49√17)⁴ - 104   +   0.000000000072... (R4)
e^{π √115} = 48³(785+351√5)³ + 744   +   0.00000000046... (H2)
e^{π √123} = 480³(32+5√41)²(8+√41)³ + 744   -   0.00000000014... (H2)
e^{π √130} = 12⁴(323+40√65)⁴ - 104   -   0.0000000000012... (R4)
e^{π √142} = 12⁴(467539+330600√2)² - 104   +   0.00000000000024... (R4)
e^{π √148} = 60³(2837+468√37)³ - 744   +   0.0000000000049... (H2)
e^{π √163} = 640320³ + 744   -   0.00000000000075... (H1)
e^{π √187} = 240³(3451+837√17)³ + 744   +   0.000000000000043... (H2)
e^{π √190} = (12√19)⁴(481+340√2)⁴ - 104   +   0.00000000000000068... (R4)
e^{π √232} = 30³(140989+26163√29)³ - 744   +   0.00000000000000032... (H2)
e^{π √235} = 528³(8875+3969√5)³ + 744   +   0.00000000000000023... (H2)
e^{π √267} = 240³(500+53√89)²(625+53√89)³   +   744 - 0.000000000000000010... (H2)
e^{π √403} = 240³(2809615+779247√13)³ + 744   +   0.000000000000000000000080... (H2)
e^{π √427} = 5280³(236674+30303√61)³ + 744   +   0.000000000000000000000012... (H2)

The little labels in parentheses identify the type: H for Hilbert class polynomial; W for Weber class polynomial; R for Ramanujan class polynomial; with a number giving the class.

A highly composite number.

Factorials

24 is a factorial: the product of consecutive integers starting with 1: 24 = 1×2×3×4. This is written "4!". The factorials are: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, ... (OEIS sequence A0142).

The factorial can be expanded to real numbers in general (and even imaginary and complex numbers) by using the Gamma function.

The 24^th pyramidal number, 1²+2²+3²+...23²+24² = 24(24+1)(2×24+1), is a perfect square, the only such case (more at 4900).

The optimal sphere packing in 4 dimensions is a "lattice packing" consisting of a repeating pattern of spheres (or "hyperspheres") attained by placing sphere centres at any location (a,b,c,d) such that all 4 coordinate are odd or all four are even. In this solution, each sphere touches 24 others. The centres of those 24 spheres are positioned at the vertices of a 4-dimensional solid called the 24-cell.

The 24-cell has 24 vertices positioned like the 16 vertices of a tesseract (4-dimensional hypercube) combined with the 8 vertices of a 16-cell that is suitably scaled (so that all 24 octahedral facets has each of its 6 vertices in a flat hyperplane). Its closest three-dimensional analogue is the rhombic dodecahedron; see 14.

There is also an optimal sphere packing in 24 dimensions, with coordinates given by the Leech lattice. It is the largest dimension for which such an optimal packing is known; each sphere touches 196560 others. Apart from being the number of dimensions, 24 is also related to the Leech lattice by the aforementioned 4900 property, and to the Monster group.

According to [143] and [191], the Umbu-Ungu language (of a small population in Papua New Guinea) uses base 24 numerals: 2 is talu, 3 is yepoko, 24 is tokapu, 48 is tokapu talu, 72 is tokapu yepoko, and 24²=576 is tokapu tokapu.

24.8412024

Approximate length of time (in hours) between successive moonrises as seen from Earth. This is 24×S/(S-1), where S is the length of the mean synodic month in mean solar days.

Squares

25 is a "perfect square", the product of an integer with itself, 5 times 5. It is called a "square" because 25 things can be arranged in a square pattern using 5 rows with 5 objects in each row.

The Pythagorean Theorem states that if a triangle has a right angle, and the side opposite the angle is length C, then A² + B² = C². 25 is a square, 5², and is the sum of two squares: 3² + 4² = 5². This is the simplest example of a square of an integer which is the sum of two other integer squares and corresponds to the 3-4-5 right triangle, which is used as an example in most ancient texts describing the Pythagorean theorem. There are many "Pythagorean triples" such as 3, 4, and 5. (See also 97 for a discussion of "Pythagorean primes".)

25 is also a square that is the sum of two consecutive squares. There aren't many of these; see 841 for more. See also 216 and 143.

25 and 27 are the only case of a square and a cube separated by 2. I have seen a proof of this but it is difficult to understand, so I cannot confirm it.

25 is the smallest Friedman number; the next is 121.

The smallest number with a persistence of 2: 2×5 = 10; 1×0 = 0.

25₁₀=31{8}. Some calculators have labeled "OCT" for "octal" and "DEC" for "decimal", inviting the following nerdy joke:

Q. Why do programmers mistake Halloween for Christmas?
A. Because 31 OCT equals 25 DEC.

See also 69.

25.4

The number of millimeters per inch (see 2.54).

The number of dimensions in the only consistent formulation of bosonic string theory. This was discovered by Lovelace in 1971, but "supersymmetric" string theories (see 11) were developed the same year, and they have prevailed over all others. See also 4900.

The number of sporadic finite simple groups (one of which is the Monster group) unless the Tits group is counted, in which case it would be 27).

As the number of letters in the English (and several other) language alphabets, 26 and 36=26+10 and its powers 26²=676, 26³=17576, etc. appear in various everyday combinatoric situations. For example, if all license plates have 2 letters followed by 4 digits, there are 26²×10000=6760000 possibilities.

This number has special divine significance in gematria because it is the sum of the letter-values of the Hebrew biblical name of God: Yod + Heh + Vau + Heh = 10 + 5 + 6 + 5 = 26.

The third cube, 3³.

27!+1 is prime. There aren't many numbers N such that N!+1 is prime; and it's hard to find them because factorials are so large. As of June 2010, the only known values of N were: 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ... (Sloane's A2981).

27×2²⁷+2²⁷+27 = 3758096411, which is prime.

27 items can be arranged into a hexagon-like arrangement with alternating sides of 3 and 4 items. So, I call it the "3¹/₂^th hexagonal number":

o o o o o o o o o o o o o o o o o o o o o o o o o o o

Randall Munroe helpfully points out that 3×9 = 3×√81 = 3)81 = 27 (but knowing this fact might do you more harm than good on an exam, see [229]).

Because 27 is the smallest factor of 999 that is not also a factor of 9 or 99, 27 is the smallest number whose reciprocal has a 3-digit repeating pattern. The next is 37. Because 27×37=999, we also have the nice relationship 1/27 = .037037037..., and 1/37 = .027027027... See also 239 and 757.

The digits of 27, 2 and 7, plus the numbers in between, add up to 27: 2+3+4+5+6+7=27. The only other 2-digit number that shares this property is 15. The next in the sequence is 429; see its entry for more.

To test a number for divisibility by 27:

Take the digits of the number in groups of 3 starting from the right, and add the resulting numbers together. If the result is more than 3 digits, repeat this process.
Check the resulting 3-digit number for divisibility by 9. If it isn't, the original number isn't divisible by 27.
If it is, divide it by 9 (see the 9 entry for a simple way to do this). Then check the answer for divisibility by 3. If it is, the original number is divisible by 27, otherwise it isn't.

27³=19683 and 1+9+6+8+3=27. The smallest number with this property is 8, and it is perhaps of interest that 8 and 27 are themselves cubes. (See 19683 for more).

In Hindu astrology there are 27 nakshatra, (which means "stars"), each controls a section of the zodiac that is 13^o20' wide; the first few are called ashwini, bharani, krittika. 27 is the closest integer approximation to the sidereal month, although most cultures with lunar Zodiac divisions tend to use 28 instead. It is also a rather nice coincidence that the size of the sun and moon in the sky (about half a degree, or 1/720 of the full circle) is almost exactly 1/27 of the distance covered by the moon each day — in other words, the fraction of the moon's size in proportion to a nakshatra is equal to the fraction of one nakshatra in proportion to the entire zodiac.

There are 27 different ways that three people can throw a choice in three-player rock-paper-scissors²³. In this variant by Paul Hsieh, the rules are a little more elaborate. As in normal rock-paper-scissors²⁴, the hand gestures are the closed fist (rock), all fingers out flat (paper) and two fingers out like a "victory" or "peace" sign (scissors). If all three players throw the same thing (a dead draw) or if they throw all three different signs (a loop or vicious circle) there is no winner and they all play again; there are 9 ways this can happen. If two players throw the same sign, the third player is the odd one out. If the odd one's sign beats the other two players' sign (a slam) then the odd one wins instantly (there are 9 ways this can happen). Otherwise, (the last 9 possibilities) the odd one's sign loses to the other two (a wipeout); the odd one is eliminated and the other two players compete using a normal two-player match.

There are also 27 gambits in Professional Rock-Paper-Scissors²⁵; a gambit is a sequence of three consecutive throws, used as a component unit of a match. (Players go so fast that they need to practice sequences of throws many times and often use memorised gambits as a way to gain the upper hand against less-experienced players who can't keep up with the pace.)

27 is a psychologically random number, similar to 17 and 37 and having no particular cultural origin. Like 37, it is often used when some random-sounding large number is needed. For example, in Graham Greene's 1953 play The Living Room one finds the line:

ROSE Since my last confession three weeks ago I've committed adultery twenty-seven times.

(27 is also my favourite cult number, for various reasons, for example it's the street number of a house where I grew up, and my age was 27 years + 27 days when I met a certain close friend.)

27.212220817

(the mean draconic month)

Length of the mean draconic (or nodical) month, in mean solar days. The draconic month is the amount of time for the Moon to complete one orbit around the Earth, measured between rising node crossings; the rising node is when the Moon crosses the plane of Earth's orbit. This period is significant because it determines when eclipses are possible. This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000003833 each year.

27.321582241

(the mean tropical month)

Length of the mean tropical month, in mean solar days. The tropical month is the amount of time for the Moon to complete one orbit around the Earth in relation to the equinox and solstice points (which shift relative to the stars because of precession). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000001506 each year.

27.321661547

(the mean sidereal month)

Length of the mean sidereal month, in mean solar days. The sidereal month is the amount of time for the Moon to complete one orbit around the Earth in relation to the stars (and thus, Zodiac signs). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000001857 each year.

27.554549878

(the mean anomalistic month)

Length of the mean anomalistic month, in mean solar days. The anomalistic month is the amount of time for the Moon to complete one orbit around the Earth, measured from perigee to perigee. The alignment of this cycle with other cycles (most notably the synodic month) affects the timing of the moon's phases (see here for more). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure decreases by 0.000000010390 each year.

A perfect number, one that is exactly half the sum of its divisors: 28=(1+2+4+7+14+28)/2. The perfect numbers are: 6, 28, 496, 8128, 33550336, ... (Sloane's A0396). See here for a complete list. See here for more about perfect numbers.

28 is used as an approximation to the number of days in the lunar month, in those cultures that need such a thing. For example, in Chinese astrology there are 28 "mansions" in the zodiac, analogous to the 12 houses of the sun (called jiao, kàng, di, fáng, etc.). In old Arabian astrology they are called manazils (the first few are alnath, albotain, azoraya). In some versions of Hindu/Indian astrology these 28 mansions are called nakshatra (Other versions have 27 nakshatra; the versions with 28 interpolate the mansion abhijit between uttarashadha and shravana).

28 also shows up in calendars as the number of years before dates repeat on the same day of the week: 28=4×7 — 4 for the number of years in the leap year cycle, 7 for the number of days in a week.

29 is the lowest base with 'easy' divisibility tests for 12 different numbers, assuming that the casting out 11's method is allowed. In base 29 you can test for divisibility by 2, 3, 4, 5, 6, 7, 10, 14, 15, 28, 29 and 30.

Here is a list of record-setters for this property (bases with high numbers of testable divisors). The divisors in bold are tested just by looking at the last digit; the divisors in the plain font are tested by the digit-addition technique (casting out 9's), and the divisors in italic are tested by alternate addition and subtraction (see the entry on 11 for a description):

base	N	divisors
2	2	2, 3
3	3	2, 3, 4
4	4	2, 3, 4, 5
5	5	2, 3, 4, 5, 6
7	6	2, 3, 4, 6, 7, 8
9	7	2, 3, 4, 5, 8, 9, 10
11	8	2, 3, 4, 5, 6, 10, 11, 12
15	9	2, 3, 4, 5, 7, 8, 14, 15, 16
19	10	2, 3, 4, 5, 6, 9, 10, 18, 19, 20
25	11	2, 3, 4, 5, 6, 8, 12, 13, 24, 25, 26
29	12	2, 3, 4, 5, 6, 7, 10, 14, 15, 28, 29, 30
35	13	2, 3, 4, 5, 6, 7, 9, 12, 17, 18, 34, 35, 36
41	14	2, 3, 4, 5, 6, 7, 8, 10, 14, 20, 21, 40, 41, 42
49	15	2, 3, 4, 5, 6, 7, 8, 10, 12, 16, 24, 25, 48, 49, 50
55	16	2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 27, 28, 54, 55, 56
71	18	2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 18, 24, 35, 36, 70, 71, 72
119	20	2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 17, 20, 24, 30, 40, 59, 60, 118, 119, 120

If you want the record-setters for prime divisors only, check the entry for 14. If you don't think the casting out 11's method should count, see the entry for 21.

29.530588235... = 25101/850

The best approximation to the length of the synodic month that can be had with an integer fraction. Better approximations are of little use because the moon is slowing down in its orbit, and before the approximation drifts enough to matter, the moon will have slowed down enough to make the approximation obsolete.

29.530588853

(the mean synodic month)

Length of the mean synodic month, in mean solar days. The synodic month is the amount of time for the Moon to complete one orbit around the Earth, measured from new moon to new moon and given in units of solar days — the common "lunar month" used for calendar purposes. This is the value for the year 2000, it increases by an average of 0.000000002162 per year. Because of the effects of Tidal acceleration with the Earth and lesser influences by other planets, the Moon's orbit is continually changing.