| Notable Properties of Specific Numbers |
Back to page 6 . . . Forward to page 8
The number that most Americans know best from its use as a speed limit also happens to be a Fibonacci number and the sum of the numbers 1 through 10 (which makes it the 10th triangular number). There are no larger numbers that are both a Fibonacci and triangular.
55 is also 52+42+32+22+12. Such numbers are called pyramidal numbers, because you can stack the squares on top of each other (with the largest on the bottom) to make a pyramid. Sometimes they are called square pyramidal numbers to distinguish them from the tetrahedral numbers. You can add the next square to each pyramidal number to get the next pyramidal number. The pyramidal numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, ... (Sloane's A000330; my MCS1936).
Another way to express the sum of squares above is: 5×5+4×4+3×3+2×2+1×1. If you change the + and × in that expression to × and exponents respectively, you get 86400000, the number of milliseconds in a day.
56 is one of the partition numbers, which count the number of ways of placing N indistinguishable balls into N indistinguishable urns (but with no empty urns). For example:
1: (*)
2: (**) (*)(*)
3: (***) (**)(*) (*)(*)(*)
5: (****) (***)(*) (**)(**) (**)(*)(*)
(*)(*)(*)(*)
7: (*****) (****)(*) (***)(**)
(***)(*)(*) (**)(**)(*) (**)(*)(*)(*)
(*)(*)(*)(*)(*)
11: (******) (*****)(*) (****)(**)
(****)(*)(*) (***)(***) (***)(**)(*)
(***)(*)(*)(*) (**)(**)(**) (**)(**)(*)(*)
(**)(*)(*)(*)(*) (*)(*)(*)(*)(*)(*)
The sequence runs: 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, ... (Sloane's A000041, more terms here).
If the urns are distinguishable, then we get the powers of two. If the balls are distinguishable, we get the Bell numbers.
There are 161280 ways to arrange the digits 1 through 5 in a 5×5 square so that each digit appears once in each row and once in each column. Such an arrangement is called a latin square. If the first row has its digits in increasing order, it is called a reduced latin square. There are 56 5×5 reduced latin squares. Here is an example:
1 2 3 4 5
2 3 5 1 4
3 5 4 2 1
4 1 2 5 3
5 4 1 3 2
9×9 latin squares are familiar to anyone who has played sudoku.
This is the value of x for which xx=10100. It is therefore the biggest number you can raise to its own power on most scientific calculators22.
57 can be expressed as a sum xy + yx where x and y are integers greater than 1 — in this case, 25+52=57. Such numbers are called Leyland numbers; the sequence begins: 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, ... (Sloane's A076980. Here is a table showing the 22 Leyland numbers less than 10000:
| x=2 | x=3 | x=4 | x=5 | ||
| y=2 | 8 | ||||
| y=3 | 17 | 54 | |||
| y=4 | 32 | 145 | 512 | ||
| y=5 | 57 | 368 | 1649 | 6250 | |
| y=6 | 100 | 945 | 5392 | ||
| y=7 | 177 | 2530 | |||
| y=8 | 320 | 7073 | |||
| y=9 | 593 | ||||
| y=10 | 1124 | ||||
| y=11 | 2169 | ||||
| y=12 | 4240 | ||||
| y=13 | 8361 |
|
See also 5.19344...×1015070.
58 is the sum of the first 7 primes: 2+3+5+7+11+13+17=58. The sequence you get by starting with 2 and adding each prime in succession is: 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, ... (Sloane's A007504). This sequence grows a little faster than the triangular numbers: for arbitrarily high n, the nth number is about n2×log(n)/2.
The center number of the 3×3 prime magic square discussed here.
Euler asserted that there were solutions to the equation A4+B4 = C4+D4, where A<C<D<B are all positive integers. He then found the first solution, A=59 (see 635318657).
The number of minutes in an hour, and the number of seconds in a minute. Degrees of angle are also divided into 60 minutes, 60×60 seconds, and the divisions can proceed further to 60×60×60 "thirds" (subdivisions of the third order)14. These subdivisions into 60's are the most familar relics of the Sumerian base-60 number system (which was really closer to being a system of using base-10 and base-6 on alternating digits). It is widely acknowledged that 60 was used in a number system because it has lots of factors; in fact it sets a record for having more factors than any smaller number. Other popular ancient factor record-setters include 12 and 360; a less popular possibility is 1260, and another related to time divisions is 10080.
61 items can be arranged in a regular pattern to make a hexagon with
5 items on a side:
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 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 o o
o o o o o
Thus 61 is called the 5th "hexagonal number". One can also make
similar patterns that are almost hexagonal; see 27. I
consider the halfway ones to be significant because they are often
used in real-world applications such as cable bundles, etc. (and
also because it makes my favorite number 27 be part of the
sequence :-)
The full sequence (including the halfway ones) runs: 1, 3,
7, 12, 19, 27, 37, 48,
61, 75, 91, 108, 127, 147, 169, 192,
217, 243, 271, ... (Sloane's A077043; my
MCS1678). For integer sizes, the formula is
3N2-3N+1, and for half-integer sizes, 3(N-1/2)2.
N: 1 1.5 2 2.5 3 3.5 4
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 o o o o o o o o o o
1 o o o o o o o o o o o o o o o o o o o o o o
3 o o o o o o o o o o o o o o o o o o o o
7 o o o o o o o o o o o o o o
12 o o o o o o o o o
19 o o o o
27
37
62 is the product of two primes. This sequence begins: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, ... (Sloane's A001358). Note that squares of primes are included, but not higher powers like 8 or 27. At first it seems that these outnumber the more composite numbers, but as you get higher, the opposite ends up being true. The same sequence omitting the squares of the primes is called distinct semiprimes. See also 1679.
A generalized Woodall number: 4×24-1. Numbers of this type are discussed on a separate page. See also 648.
64 is the number of ways of distributing 7 indistinguishable balls into one or more distinguishable urns (with no empty urns). Here are examples for N=1 through 4:
1: (*)
2: (**) (*)(*)
4: (***) (**)(*) (*)(**) (*)(*)(*)
8: (****) (***)(*) (**)(**) (**)(*)(*)
(*)(***) (*)(**)(*) (*)(*)(**) (*)(*)(*)(*)
The sequence is just the powers of 2 (in this case, 2N-1). If the balls are distinguishable, we get instead the ordered Bell numbers. If the balls and urns are both indistinguishable, we get the partition numbers.
65 is the smallest number that can be expressed as the sum of two distinct squares in two different ways: 65 = 12+82 = 42+72. (If they do not need to be distinct, 50 is the smallest.) See also 1729 and 635318657.
65 is the magic constant of a 5×5
magic square. Here are two examples:
1 15 24 8 17 11 10 4 23 17
23 7 16 5 14 18 12 6 5 24
20 4 13 22 6 25 19 13 7 1
12 21 10 19 3 2 21 20 14 8
9 18 2 11 25 9 3 22 16 15
All rows and columns add to 65, as do the two diagonals, and also any other set of 5 squares that is arranged in a pattern with 180-degree rotational symmetry. (Magic squares with this property are called associative magic squares.) In addition, all the diagonals, including those that "wrap around" (like the five numbers 24, 5, 6, 12 and 18 in the left example) add to 65. This makes it a panmagic square. 5×5 is the smallest size magic square that can be both associative and panmagic.
If it is an associative magic square, any pattern of N cells with
180-degree rotational symmetry adds up to the magic constant. Most are
rather nice-looking patterns. In the 5×5 case, there are 23
patterns, ignoring rotations and reflections. They are labeled here
with unique numbers that I use to identify the pattern, and a 1, 2 or
4 telling how many more there would be if you count rotations and
reflections:
10 4 11 4 12 4 13 1 15 2 16 4
X X . . . X . X . . X . . X . X . . . X X . . . . X . . . .
. . . . . . . . . . . . . . . . . . . . . X . . . . . X . .
. . X . . . . X . . . . X . . . . X . . . . X . . . . X . .
. . . . . . . . . . . . . . . . . . . . . . . X . . . X . .
. . . X X . . X . X . X . . X X . . . X . . . . X . . . . X
17 2 20 4 21 2 23 2 24 4 25 4
X . . . . . X X . . . X . X . . X . . . . X . . . . X . . .
. . . X . . . . . . . . . . . X . . . . . X . . . . . X . .
. . X . . . . X . . . . X . . . . X . . . . X . . . . X . .
. X . . . . . . . . . . . . . . . . . X . . . X . . . X . .
. . . . X . . X X . . X . X . . . . X . . . . X . . . . X .
26 4 27 2 28 4 29 4 33 4 34 2
. X . . . . X . . . . X . . . . X . . . . . X . . . . X . .
. . . X . . . . . X . . . . . . . . . . . X . . . . . X . .
. . X . . . . X . . X . X . X . X X X . . . X . . . . X . .
. X . . . X . . . . . . . . . . . . . . . . . X . . . X . .
. . . X . . . . X . . . . X . . . . X . . . X . . . . X . .
37 1 38 2 70 4 71 1 83 1
. . X . . . . X . . . . . . . . . . . . . . . . .
. . . . . . . . . . . X X . . . X . X . . . X . .
X . X . X . X X X . . . X . . . . X . . . X X X .
. . . . . . . . . . . . X X . . X . X . . . X . .
. . X . . . . X . . . . . . . . . . . . . . . . .
More generally, one could refer to a "pattern" as any set of N numbers that adds up to the magic constant. In the 5×5 case, there are 1394 such sets. However, most of them do not have a nice geometrical layout. For example, consider: (3, 4, 15, 20, 23). These 5 numbers add up to 65, but if you find them in either of the squares above, you'll see they're in a sort of random pattern.
In the two sample magic squares, the left example is constructed by the following method. A similar method works for all N×N magic squares where N is odd:
1. Start in the center square, with the value (N2+1)/2.
2. Move one up; if this takes you off the top go to the bottom
of the same column.
3. Move two to the right; if this takes you off the right edge continue
at the left end of the same row.
4. Write the next number in sequence in this square (if the next number
is N2+1, go back to 1)
5. Repeat steps 2-4 another N-1 times.
6. Now move one down and one to the right; as before if this takes you
off the right continue at the left or if it takes you off the bottom
continue at the top of the same column.
7. Write the next number in sequence in this square.
8. Repeat steps 2 through 7 until you have filled all the squares.
The right example is constructed by a similar method, but in steps 2-3 moving one down and one to the right and in step 6 moving one to the left.
66 is the lowest base with 'easy' divisibility tests for 5 different primes, assuming that the casting out 11's technique is not considered 'easy'.
In base 10, there are well-known tests for divisibility by the primes 2, 3, and 5, and by the composite numbers 9 and 10. (The tests for 2, 5 and 10 look at the last digit; the tests for 3 and 9 use the repeated digit addition technique.) Are there bases for which there are more numbers that can be easily tested? In base 9, you can test for divisibility by 3 and 9 by looking at the last digit, and by 2, 4, or 8 by using the digit-addition trick. So base 9 has just as many easy tests (5) as base 10. But the number of primes is less (just two: 2 and 3). In base 6, you can test for divisibility by the same three primes (2, 3, 5) as in base 10. Since division by composites can always be broken down into division by multiple primes, what really matters the most, for convenience and utility, is to have tests for as many primes as possible. Base 66 is the smallest base for which you can easily test for divisibility by 5 different primes: 2, 3, 5, 11 and 13. Notice they aren't the 5 smallest primes (7 is missing).
Here is a table of the record-setters for this property. The primes in bold are the ones tested by looking at the last digit, the others require the digit-addition technique:
| base | N | prime divisors |
| 2 | 1 | 2 |
| 3 | 2 | 2, 3 |
| 6 | 3 | 2, 3, 5 |
| 15 | 4 | 2, 3, 5, 7 |
| 66 | 5 | 2, 3, 5, 11, 13 |
| 210 | 6 | 2, 3, 5, 7, 11, 19 |
| 715 | 7 | 2, 3, 5, 7, 11, 13, 17 |
| 7315 | 8 | 2, 3, 5, 7, 11, 19, 23, 53 |
| 38571 | 9 | 2, 3, 5, 7, 13, 19, 23, 29, 43 |
| 254541 | 10 | 2, 3, 5, 7, 11, 13, 17, 23, 31, 89 |
| 728365 | 11 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 41 |
| 11243155 | 12 | 2, 3, 5, 7, 11, 13, 17, 19, 29, 53, 61, 139 |
| 58524466 | 13 | 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 41, 73, 97 |
| 812646121 | 14 | 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 43, 47, 71, 109 |
| 5163068911 | 15 | 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 67 |
Note that if you want the record-setters for just the last-digit test technique, they are the same as the primorials.
If you count the casting out 11's method as an easy factorization test, then you get the table shown under the 14 entry. If you want the record-setters for all divisors (not just primes), see the entry for 21 (or 29 if the casting out 11 method is included).
67 and 71 are both prime, and differ by four. Because of this, they are called cousin primes (in an analogy to twin primes). The first few pairs of cousin primes are at: 3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, ... (Sloane's A023200). The odd number in between (in this case 69) is always divisible by 3, except in the special case of the cousin primes 3 and 7.
68 is a member of an iteratively-defined sequence called the Perrin (or Ondrej Such) numbers. It is defined in a similar way to the Fibonacci and Pell numbers: A0 = 0; A1 = 2; A2 = 3; AN+1 = AN-1 + AN-2. The sequence starts: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367, ... (Sloane's A001608; my MCS197664). Because it has the same iteration rule as the Padovan numbers, both have the same asymtotic ratio, about 1.3247.
Another cult number, notable (among other reasons) for being the same when turned upside down. Such numbers are called strobogrammatic. The last three years with this property were 1691, 1881 and 1961; the next isn't until 6009. The complete sequence of invertable numbers starts: 0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, ... (Sloane's A000787). To make an invertible number, start with any combination of the digits 0, 1, 6, 8 and 9; then add the same set of digits in reverse order and inverted. If the middle two digits are 00, 11 or 88, you can make another invertible number by removing one of the duplicated digts. For example, starting with 68 we can make the two invertable numbers 6889 and 689. In 1961 there was actually an article in a mainstream math journal titled "Strobogrammatic Years".
The Jargon File73 entry for 69 points out that 6916=10510 and 6910=1058. 25 also has this property. See also 69105.
Many have found that 69 is the largest value of which you can calculate the factorial on a scientific calculator: 69! ≅ 1.711×1098, and 70! overflows. Here's something you might find even cooler: Put in 69, then square it 5 times in a row: ((((692)2)2)2)2 ≅ 6.969×1058.
The Pell numbers are similar to the Fibonacci numbers and are generated by the formula An = 2 An-1 + An-2. The sequence runs: 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, ... (Sloane's A000129; my MCS1684). The ratio of successive terms approaches 1 plus the square root of 2.
70 appears in many places throughout the Hebrew Bible (Gen 46:27, 50:3; Exo 1:5, 15:27, 24:1; Num 7:13 etc., 11:16 etc., 33:9; Deu 10:22; Jud 1:7, 8:30 etc.; 2Ki 10:1; and so on)
Back to page 6 . . . Forward to page 8