Whole Numbers are positive numbers, including 0, with no decimal or fractional parts
Since 191 ≥ 0 and it is an integer, 191 is a whole number
Prime or Composite Test: Since 191 is only divisible by 1 and itself, it is a prime number
Perfect/Deficient/Abundant Test: A perfect number is a number who has a divisor sum equal to itself. An abundant number is a number who has a divisor sum greater than the number, otherwise, it is deficient. Divisor Sum = 1 Since our divisor sum of 1 < 191, 191 is a deficient number!
Odd or Even Test (Parity Function): A number is even if it is divisible by 2, else it is odd
95.5 =
191
2
Since 95.5 is not an integer, 191 is not divisible by 2, and therefore, it is an odd number This can be written as A(191) = Odd
Evil or Odious Test: A number is evil is there are an even number of 1's in the binary expansion, else, it is odious. Using our decimal to binary calculator, we see the binary expansion of 191 is 10111111 Since there are 7 1's in the binary expansion which is an odd number, 191 is an odious number
Triangular Test: A number is triangular if it can be stacked in a pyramid with each row above containing one item less than the row before it, ending with 1 item at the top Using a bottom row of 20 items, we cannot form a pyramid with our numbers, therefore 191 is not triangular
Rectangular Test: A number n is rectangular is if there is an integer m such that n = m(m + 1) No integer m exists such that m(m + 1) = 191, therefore 191 is not rectangular
Automorphic (Curious) Test: A number (n) is automorphic (curious) if n2 ends with n 1912 = 191 x 191 = 36481 Since 36481 does not end with 191, it is not automorphic (curious)
Undulating Test: A number (n) is undulating if the digits that comprise n alternate in the form abab In this case, a = 1 and b = 9 In order to be undulating, Digit 1: 111 should be equal to 1 In order to be undulating, Digit 2: 999 should be equal to 9 In order to be undulating, Digit 3: 111 should be equal to 1 Since all 3 digits form our abab undulation pattern, 191 is undulating
Square Test: A number (n) is a square if there exists a number m such that m2 = n Analyzing squares, we see that 132 = 169 and 142 = 196 which do not equal 191 Therefore, 191 is not a square
Cube Test: A number (n) is a cube if there exists a number m such that m3 = n Analyzing cubes, we see that 53 = 125 and 63 = 216 which do not equal 191 Therefore, 191 is not a cube
Palindrome Test: A number (n) is a palindrome if the number read backwards equals the number itself The number read backwards is 191 Since 191 is the same backwards and forwards, it is a palindrome
Palindromic Prime Test: A number is a palindromic prime if it is both prime and a palindrome From above, since 191 is both prime and a palindrome, it is a palindromic prime
Repunit Test: A number is repunit if every digit is equal to 1 Since there is at least one digit in 191 not equal to 1, then it is NOT repunit
Apocalyptic Power Test: A number (n) is apocalyptic power if 2n contains the consecutive digits 666. 2191 = 3.1385508676933E+57 Since 2191 does not contain the sequence 666, 191 is NOT an apocalyptic power
Pentagonal Test: A pentagonal number is one which satisfies the form:
n(3n - 1)
2
Check values of 11 and 12 Using n = 12, we have:
12(3(12 - 1)
2
12(36 - 1)
2
12(35)
2
420
2
210 ← Since this does not equal 191, this is NOT a pentagonal number
Using n = 11, we have:
11(3(11 - 1)
2
11(33 - 1)
2
11(32)
2
352
2
176 ← Since this does not equal 191, this is NOT a pentagonal number
Hexagonal Test: A number n is hexagonal is if there is an integer m such that n = m(2m - 1) No integer m exists such that m(2m - 1) = 191, therefore 191 is not hexagonal
Octagonal Test: A number n is octagonal is if there is an integer m such that n = m(3m - 3) No integer m exists such that m(3m - 2) = 191, therefore 191 is not octagonal
Tetrahedral (Pyramidal) Test: A tetrahedral (Pyramidal) number is one which satisfies the form:
n(n + 1)(n + 2)
6
Check values of 9 and 10 Using n = 10, we have:
10(10 + 1)(10 + 2)
6
10(11)(12)
6
1320
6
220 ← Since this does not equal 191, this is NOT a tetrahedral (Pyramidal) number
Using n = 9, we have:
9(9 + 1)(9 + 2)
6
9(10)(11)
6
990
6
165 ← Since this does not equal 191, this is NOT a tetrahedral (Pyramidal) number
Narcissistic (Plus Perfect) Test: An m digit number n is narcissistic if it is equal to the square sum of it's m-th powers of its digits 191 is a 3 digit number, so m = 3 Square sum of digitsm = 13 + 93 + 13 Square sum of digitsm = 1 + 729 + 1 Square sum of digitsm = 731 Since 731 <> 191, 191 is NOT narcissistic (plus perfect)
Catalan Test: The nth Catalan number Cn is denoted by:
