GOTW- Counting - Page 8

Sixty-five is the 23rd^[1] semiprime and the 3rd of the form (5.q)it is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0.

Sixty-five has an aliquot sum of 19 and is the first composite number in the 19-aliquot tree.

This number is the magic constant of 5 by 5 normal magic square:

This number is also the magic constant of n-Queens Problem for n = 5.^[2]

65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two ways, 65 = 8² + 1² = 7² + 4².

It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).

There are only 65 known Euler's idoneal numbers.

65=1⁵+2⁴+3³+4²+5¹

• 66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number. Being a multiple of a perfect number, 66 is itself a semiperfect number.
• Since it is possible to find sequences of 66 consecutive integers such that each inner member shares a factor with either the first or the last member, 66 is an Erdos–Woods number.

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Sixty-seven is the 19th prime number (the next is 71), an irregular prime, a lucky prime, the sum of five consecutive primes (7 + 11 + 13 + 17 + 19), and a Heegner number.

Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of 18, 67 is a Pillai prime.

In a Voronoi diagram created using points from the prime spiral, no prime less than 10242 will have a rounder Voronoi cell than 67.

The aliquot sum of sixty-nine is 27 within the aliquot sequence (69,27,13,1,0) 69 being the third composite number in the 13-aliquot tree.

69 is a semiprime. Furthermore, since the two factors of 69 are both Gaussian primes, 69 is a Blum integer.

Adding up the divisors of 1 through 9 gives 69.

Because 69 has an odd number of 1s in its binary representation, it is sometimes called an "odious number." Of note is that 69 (4 761) and 69 (328 509) together use every decimal digit from 0-9. 69 is equal to 105 octal, while 105 is equal to 69 hexadecimal. This same property can be applied to all numbers from 64 to 69.

On many handheld scientific and graphing calculators, the highest factorial that can be calculated, due to memory limitations, is 69 or 1.71122452410⁹⁸.

The number 69 written in Western Arabic numeral notation can be rotated 180 and remain visually the same.

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Its factorization makes it a sphenic number. 70 is a Pell number and a generalized heptagonal number, one of only two numbers to be both.^[1] Also, it is the seventh pentagonal number and the fourth 13-gonal number, as well as the fifth pentatope number. It is the smallest weird number.

Since it is possible to find sequences of 70 consecutive integers such that each inner member shares a factor with either the first or the last member, 70 is an Erdos–Woods number.

In base 10, it is a Harshad number.

Its factorization makes it a sphenic number. 70 is a Pell number and a generalized heptagonal number, one of only two numbers to be both.^[1] Also, it is the seventh pentagonal number and the fourth 13-gonal number, as well as the fifth pentatope number. It is the smallest weird number.

Since it is possible to find sequences of 70 consecutive integers such that each inner member shares a factor with either the first or the last member, 70 is an Erdos–Woods number.

In base 10, it is a Harshad number.