### About the Number 169 (one hundred sixty-nine)

MathBabbler posted the nBAB for the square-number 196 on 21 July 2009 and he noticed there were nBABs for all square-numbers less than 196 expected for `169`.

```   MathBabbler Number Analyst (MBNA) output:
=========================================
169 is a natural, whole, integer
169 is odd
169 proper divisors are: 1,13,
169 is deficient (sum of divisors is 14)
169 is unhappy
169 is not a Harshad number
169 is not prime
169 has the prime factors: 13*13 (sum=26)
169 is semiprime (biprime or 2-almost prime)
169 sum of prime factors is: 26
169 is a lucky number
169 in octal is 0251
169 in hexadecimal is 0xa9
169 in binary is 10101001 (is evil)
169 nearest square numbers: 169 is square (169 is square)
sqrt(169) = 13
ln(169) = 5.1299
log(169) = 2.22789
169 is 13^2
169 reciprocal is .00591715976331360946745562130177
169! is 4.26907e+304
169 is 53.7944 Pi years
169 is 8 score and 9 years
169 written as a Roman numeral is CLXIX
```

The MBNA's basen program produced the following output for the number `169`.

```   169 base16 in base10 is 361
169 base15 in base10 is 324
169 base14 in base10 is 289
169 base13 in base10 is 256
169 base12 in base10 is 225
169 base11 in base10 is 196
```

Observe... 196, 225, 256, 289, 324, 361 are all square numbers.

The basen program was re-executed to see if the aforementioned pattern continued in bases greater than 16 and it does.

```   169 base60 in base10 is 3969
169 base59 in base10 is 3844
169 base58 in base10 is 3721
169 base57 in base10 is 3600
169 base56 in base10 is 3481
....
```

Observe... 3481, 3600, 3271, 3844, 3969 are square numbers.

The basen program was re-executed using 144, 121, and 100 as inputs. Observe the outputs...

```   144 base16 in base10 is 324
144 base15 in base10 is 289
144 base14 in base10 is 256
144 base13 in base10 is 225
144 base12 in base10 is 196
144 base11 in base10 is 169
144 base9 in base10 is 121
144 base8 in base10 is 100
144 base7 in base10 is 81
144 base6 in base10 is 64
144 base5 in base10 is 49

121 base16 in base10 is 289
121 base15 in base10 is 256
121 base14 in base10 is 225
121 base13 in base10 is 196
121 base12 in base10 is 169
121 base11 in base10 is 144
121 base9 in base10 is 100
121 base8 in base10 is 81
121 base7 in base10 is 64
121 base6 in base10 is 49
121 base5 in base10 is 36
121 base4 in base10 is 25
121 base3 in base10 is 16

100 base16 in base10 is 256
100 base15 in base10 is 225
100 base14 in base10 is 196
100 base13 in base10 is 169
100 base12 in base10 is 144
100 base11 in base10 is 121
100 base9 in base10 is 81
100 base8 in base10 is 64
100 base7 in base10 is 49
100 base6 in base10 is 36
100 base5 in base10 is 25
100 base4 in base10 is 16
100 base3 in base10 is 9
100 base2 in base10 is 4
```

NumberGossip.com reported that `169` is the "smallest square that is prime when turned upside down (691)." MathBabbler did not know what it meant to turn a number upside down.

Wikipedia.org reported that `169` is the "sum of seven consecutive primes: 13 + 17 + 19 + 23 + 29 + 31 + 37.

Wikipedia.org reported that `169` is 13*13, while 31*31 is the square number 961.

This nBAB was created using a version of the MBNA that didn't check for powerful numbers. `169` is a powerful number.

A powerful number is a "positive integer m that for every prime number p dividing m, p^2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a^2b^3, where a and b are positive integers." Note: Powerful numbers are also called squarefull, square-full or 2-full numbers.

OEIS.org::id:A001694

```   1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100,
108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243,
256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484,
500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800,
841, 864, 900, 961, 968, 972, 1000, ...
```

Note: This was not the first nBAB to document powerful numbers.

Creator: Gerald Thurman [gthurman@gmail.com]
Created: 22 July 2009 