Template:Sequence of the Day for September 6

Intended for: September 6, 2013

Timetable

• First draft entered by Alonso del Arte on September 3, 2011 ✓
• Draft reviewed by Daniel Forgues on September 5, 2011 ✓
• Draft to be approved by August 6, 2011

Yesterday's SOTD * Tomorrow's SOTD

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

---

A066888: Number of primes between the

n

th triangular number (exclusive) and the next (inclusive):

π (Tn +1 )  −  π (Tn  ), n   ≥   0

.

{ 0, 2, 1, 1, 2, 2, 1, 2, 3, 2, ... }

Triangle begins

 1               (0 prime in (T_0, T_1])

 2  3            (2 primes in (T_1, T_2])

 4  5  6         (1 prime in (T_2, T_3])

 7  8  9 10      (1 prime in (T_3, T_4])

11 12 13 14 15   (2 primes in (T_4, T_5])

It is believed that the initial zero of this sequence is the only zero in the sequence. That is to say, that there is always a prime in

(Tn , Tn +1]

for

n > 0

. (Or there is always a positive noncomposite for

n   ≥   0

, since 1 is a unit.) (With the exception of 1 and 3, all triangular numbers are composite, since

Tn = n (n + 1) 2

.)