Template:Sequence of the Day for August 3

Intended for: August 3, 2011

Timetable

• First draft entered by Alonso del Arte on May 3, 2011 ✓
• Draft reviewed by Alonso del Arte on June 3, 2011 (same person who entered first draft due to deadline running out) ✓
• Draft to be approved by July 3, 2011

Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.

---

A001248: Squares of prime numbers.

{ 4, 9, 25, 49, 121, 169, 289, ... }

Among the perfect squares (the only numbers with an odd number of divisors), the squares of primes are the only numbers with exactly three divisors (1, their square root, themselves). In doing the sieve of Eratosthenes, this sequence gives the first composite number that you’ll cross off after you identify the

n

th prime (that is, unless you like to cross off again the composite numbers that have already been crossed off for a smaller prime factor—e.g., for the third prime, 5, the first number to cross off is 25, because 10, 15 and 20 should have already been crossed off for 2, 3 and 2 respectively.

Note that since all primes, except 2 and 3 which are the prime factors of 6, are congruent to ±1 (mod 6), it means that

(  pn +1 ) 2  −  (  pn  ) 2, n   ≥   3,

is a multiple of 24. (See A069482 and A075888.) Furthermore,

(  pn  ) 2  −  1, n   ≥   3,

is also a multiple of 24. (See A024702.)