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

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