GOTW- Counting - Page 9

Seventy-two is the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). The product of 8 and 9, 72 is a pronic number.

The sum of Euler's totient function f(x) over the first fifteen integers is 72. There are 17 solutions to the equation f(x) = 72, more than any integer below 72, making it a highly totient number.

72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵ = 72⁵.

The sum of the eighth row of Lozanic's triangle is 72.

In a plane, the exterior angles of a regular pentagon measure 72 degrees each.

In base 10, the number 72 is a Harshad number.

• Seventy-three is the 21st prime number. The previous is seventy-one, with which it comprises the 8th twin prime. It is also a permutable prime with thirty-seven. 73 is a star number.
• 73 is the largest minimal Primitive root in the first 100000 primes. In other words, if p is one of the first 100000 primes, then at least one of the primes 3, 5, 7, 11, 13, 17, ..., 73 is a primitive root modulo p.
• 73 is the smallest prime congruent to 1 modulo 24.
• The mirror of 73, the 21st prime number, 37, is the 12th prime number. The number 21 includes factors 7 and 3. Seventy-three in binary, 1001001, is a palindrome. In addition, of the 7 binary digits representing 73, there are 3 ones. Also, 37+12=49 (seven squared) and 73+21=94=47*2, 47+2 also being equal to seven squared. Additionally, both 73 and its mirror, 37, are sexy primes twice over, as 31, 43, 67 and 79 are all prime numbers.
• The square root of 73 is 8.544. The sum of which (8+5+4+4) equals the product of 7 and 3; 21.
• Every positive integer can be written as the sum of 73 or fewer sixth powers (see Waring's problem).
• In base 5, the smallest prime with a composite sum of digits is 73.
• 73 is the only prime repunit in base 8 (111₈).
• 73 is the smallest factor of the first composite generalized Fermat number in base 10 (10⁴+1 = 10,001 = 73*137).

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Seventy-four is the twenty-first distinct semiprime and the eleventh of the form 2q. The aliquot sum of 74 is 40 within the aliquot sequence (74,40,43,1,0) 74 being the sixth composite number in the 43-aliquot tree.

Seventy-four is a nontotient. There are 74 different non-Hamiltonian polyhedra with a minimum number of vertices.

As the sum of the first five pentagonal numbers, seventy-five is a pentagonal pyramidal number. It is also an enneagonal number.

It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 7, 5, 12, 17, 29, 46, 75... But there is no integer that added up to its own digits adds up to 75, hence 75 is a self number.

Excluding the infinite sets, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).