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

`.`

169MathBabbler 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 CLXIXThe MBNA's

basenprogram produced the following output for the number`.`

169169 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 196Observe... 196, 225, 256, 289, 324, 361 are all square numbers.

The

basenprogram 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

basenprogram 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 4NumberGossip.com reported that

`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.`

169Wikipedia.org reported that

`is the "sum of seven consecutive primes: 13 + 17 + 19 + 23 + 29 + 31 + 37.`

169Wikipedia.org reported that

`is 13*13, while 31*31 is the square number 961.`

169This nBAB was created using a version of the MBNA that didn't check for

powerfulnumbers.`is a powerful number.`

169A

powerfulnumber 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

This work is licensed under a Creative Commons Attribution 3.0 United States License.